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Unit III Quadratic Equations 1

Section 71 ndash Solving Quadratic Equations by Graphing

Investigating Solutions to Quadratic Equations

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Indicate the positions along the quadratic trajectory above where the rocket

attains a height of 0

(b) For the function above h(t) = ndash20t2 + 120t which variable is replaced

by the height 0 for those positions

(c) Write the resulting quadratic when the variable mentioned above is

replaced by 0

t

h(t)

Goal

Solving Quadratic Equations by Graphing

Unit III Quadratic Equations 2

(d) Use graphing software (httpswwwdesmoscom) to determine the times

below where the rocket attains a height of 0

Remember

Zeros of a Quadratic Function

What is a Quadratic Equation

A second degree polynomial equation

Standard Form of a quadratic Equation ax2 + bx + c = 0

t

h(t)

x

y

Zeros of a Quadratic Function

The zero(s) of a quadratic function represent

the position(s) where the height is _________

The zero(s) of a quadratic function

are also referred to as the _________________

Solutions to Quadratic Equation

Solutions to the quadratic equation

ndash20t2 + 120t = 0 are the ________________

of the quadratic function

times where height is 0 are

t1 = _____ and t2 = ______

h(t) = ndash20t2 + 120t

Unit III Quadratic Equations 3

(e) Use the graph below to determine the times where the rocket attained a

height of 160 m

t1 = _____ t2 = _____

(f) Use the quadratic function h(t) = ndash20t2 + 120t to write the quadratic

equation that represents the height at 160 m

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash2t2 + 120t

How can we algebraically attain the time(s) to solve a quadratic

equation ax2 + bx + c = 0 such as ______________________

Unit III Quadratic Equations 4

Section 72 ndash Solving Quadratic Equations by Factoring

(I) Solving Quadratic Equations by Factoring

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

t

h(t)

20406080

100120140160180200

At a height of 160 m the quadratic function h(t) = ndash20t2 + 120t

develops into a quadratic equation

160 = ndash20t2 + 120t

We can determine the two times where the missile attains a

height of 160 m by FACTORING the quadratic equation

160 = ndash20t2 + 120t

and using the zero product property to isolate and solve for t

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 5

Solve for t by factoring

160 = ndash20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor

Example Solve by factoring

(i) ndash20x2 + 120x = 0 (ii) 12x2 = ndash8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor

Factor the remaining trinomial

To isolate t apply the Zero Product Property

If the product of two real numbers is zero ( a bull b = 0)

then one or both must be zero

In other words a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

1(a) x( x + 5 ) (b) ( 3x + 1 )( 3x ndash 1 ) (c) ( 3x ndash 1 )( x ndash 2 ) (d) (5)( x + 4 )( x ndash 1 )

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

A football is kicked and its trajectory is modeled by the function h(t) = ndash4t2 + 20t + 1 where h(t) represents height in feet ant t is time

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

7 For a quadratic function the value of b2 ndash 4ac = 8 Which is true about the graph of the quadratic

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

12 The trajectory of a missile fired from a ship is modeled by the function h(t) = ndash3t2 + 36t + 4 where

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

Example 1 Modeling Area Compression Problems The owner of a new home would like to construct a rectangular driveway

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

a) 1199092 minus 2119909 minus 4 = 0 b) 119909(119909 minus 4) = minus3 c) 21199092 minus 12119909 + 14 = 0

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

4 Determine the quadratic equation 1198861199092 + 119887119909 + 119888 = 0 that has the following roots

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 2

(d) Use graphing software (httpswwwdesmoscom) to determine the times

below where the rocket attains a height of 0

Remember

Zeros of a Quadratic Function

What is a Quadratic Equation

A second degree polynomial equation

Standard Form of a quadratic Equation ax2 + bx + c = 0

t

h(t)

x

y

Zeros of a Quadratic Function

The zero(s) of a quadratic function represent

the position(s) where the height is _________

The zero(s) of a quadratic function

are also referred to as the _________________

Solutions to Quadratic Equation

Solutions to the quadratic equation

ndash20t2 + 120t = 0 are the ________________

of the quadratic function

times where height is 0 are

t1 = _____ and t2 = ______

h(t) = ndash20t2 + 120t

Unit III Quadratic Equations 3

(e) Use the graph below to determine the times where the rocket attained a

height of 160 m

t1 = _____ t2 = _____

(f) Use the quadratic function h(t) = ndash20t2 + 120t to write the quadratic

equation that represents the height at 160 m

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash2t2 + 120t

How can we algebraically attain the time(s) to solve a quadratic

equation ax2 + bx + c = 0 such as ______________________

Unit III Quadratic Equations 4

Section 72 ndash Solving Quadratic Equations by Factoring

(I) Solving Quadratic Equations by Factoring

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

t

h(t)

20406080

100120140160180200

At a height of 160 m the quadratic function h(t) = ndash20t2 + 120t

develops into a quadratic equation

160 = ndash20t2 + 120t

We can determine the two times where the missile attains a

height of 160 m by FACTORING the quadratic equation

160 = ndash20t2 + 120t

and using the zero product property to isolate and solve for t

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 5

Solve for t by factoring

160 = ndash20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor

Example Solve by factoring

(i) ndash20x2 + 120x = 0 (ii) 12x2 = ndash8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor

Factor the remaining trinomial

To isolate t apply the Zero Product Property

If the product of two real numbers is zero ( a bull b = 0)

then one or both must be zero

In other words a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

1(a) x( x + 5 ) (b) ( 3x + 1 )( 3x ndash 1 ) (c) ( 3x ndash 1 )( x ndash 2 ) (d) (5)( x + 4 )( x ndash 1 )

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

A football is kicked and its trajectory is modeled by the function h(t) = ndash4t2 + 20t + 1 where h(t) represents height in feet ant t is time

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

7 For a quadratic function the value of b2 ndash 4ac = 8 Which is true about the graph of the quadratic

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

12 The trajectory of a missile fired from a ship is modeled by the function h(t) = ndash3t2 + 36t + 4 where

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

Example 1 Modeling Area Compression Problems The owner of a new home would like to construct a rectangular driveway

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

a) 1199092 minus 2119909 minus 4 = 0 b) 119909(119909 minus 4) = minus3 c) 21199092 minus 12119909 + 14 = 0

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

4 Determine the quadratic equation 1198861199092 + 119887119909 + 119888 = 0 that has the following roots

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 3

(e) Use the graph below to determine the times where the rocket attained a

height of 160 m

t1 = _____ t2 = _____

(f) Use the quadratic function h(t) = ndash20t2 + 120t to write the quadratic

equation that represents the height at 160 m

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash2t2 + 120t

How can we algebraically attain the time(s) to solve a quadratic

equation ax2 + bx + c = 0 such as ______________________

Unit III Quadratic Equations 4

Section 72 ndash Solving Quadratic Equations by Factoring

(I) Solving Quadratic Equations by Factoring

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

t

h(t)

20406080

100120140160180200

At a height of 160 m the quadratic function h(t) = ndash20t2 + 120t

develops into a quadratic equation

160 = ndash20t2 + 120t

We can determine the two times where the missile attains a

height of 160 m by FACTORING the quadratic equation

160 = ndash20t2 + 120t

and using the zero product property to isolate and solve for t

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 5

Solve for t by factoring

160 = ndash20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor

Example Solve by factoring

(i) ndash20x2 + 120x = 0 (ii) 12x2 = ndash8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor

Factor the remaining trinomial

To isolate t apply the Zero Product Property

If the product of two real numbers is zero ( a bull b = 0)

then one or both must be zero

In other words a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

1(a) x( x + 5 ) (b) ( 3x + 1 )( 3x ndash 1 ) (c) ( 3x ndash 1 )( x ndash 2 ) (d) (5)( x + 4 )( x ndash 1 )

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

A football is kicked and its trajectory is modeled by the function h(t) = ndash4t2 + 20t + 1 where h(t) represents height in feet ant t is time

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

7 For a quadratic function the value of b2 ndash 4ac = 8 Which is true about the graph of the quadratic

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

12 The trajectory of a missile fired from a ship is modeled by the function h(t) = ndash3t2 + 36t + 4 where

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

Example 1 Modeling Area Compression Problems The owner of a new home would like to construct a rectangular driveway

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

a) 1199092 minus 2119909 minus 4 = 0 b) 119909(119909 minus 4) = minus3 c) 21199092 minus 12119909 + 14 = 0

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

4 Determine the quadratic equation 1198861199092 + 119887119909 + 119888 = 0 that has the following roots

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 4

Section 72 ndash Solving Quadratic Equations by Factoring

(I) Solving Quadratic Equations by Factoring

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

t

h(t)

20406080

100120140160180200

At a height of 160 m the quadratic function h(t) = ndash20t2 + 120t

develops into a quadratic equation

160 = ndash20t2 + 120t

We can determine the two times where the missile attains a

height of 160 m by FACTORING the quadratic equation

160 = ndash20t2 + 120t

and using the zero product property to isolate and solve for t

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 5

Solve for t by factoring

160 = ndash20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor

Example Solve by factoring

(i) ndash20x2 + 120x = 0 (ii) 12x2 = ndash8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor

Factor the remaining trinomial

To isolate t apply the Zero Product Property

If the product of two real numbers is zero ( a bull b = 0)

then one or both must be zero

In other words a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 5

Solve for t by factoring

160 = ndash20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor

Example Solve by factoring

(i) ndash20x2 + 120x = 0 (ii) 12x2 = ndash8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor

Factor the remaining trinomial

To isolate t apply the Zero Product Property

If the product of two real numbers is zero ( a bull b = 0)

then one or both must be zero

In other words a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares

Perfect Square Numbers or Factors to remember

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example Solve by factoring

(i) 81x2 = 49 (ii) 121 ndash 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER The difference of two squares factors by pattern

a2 ndash b2 = (a ndash b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0

Example Solve by factoring

(i) x2 + 3x = 18 (ii) ndash2p2 ndash 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0

Example Solve by factoring

(i) 6x2 ndash 11x = 10 (ii) 8x2 = 22x ndash 15

P411 1 ndash 4

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 8

Section 72 ndash Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0

Example Determine the ROOTS for

(a) x(3x + 7) = 6

(b) ndash32x2 + 64x = ndash256 (c) minus1

31199092 = minus119909 + 6

bullExpress each quadratic equation in

standard form ax2 + bx + c = 0 and factor

bullSet each factor equal to zero and solve

each linear equation

Goal

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= ndash49x2 + 392x + 441 where y represents the

height in meters and x is time in seconds

Algebraically determine

(a) How long will it take for the missile to hit the ocean

(b) How long will it take for the missile to attain a height of 1225 metres

x

y Quadratic Path is modeled by

y = ndash49x2 + 392x + 441

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example

An osprey dives toward the water to catch a salmon Its height above the water in

metres t seconds after it begins its dive is approximated by 2( ) 5 30 45h t t t

Algebraically determine the time it takes for the osprey to reach a return height of

20 m

Example

A travel agency has 16 people signed up for a trip The revenue for the trip is

modeled by the function

R(x) = ndash100x2 + 800x + 38400

where x represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $40000

P411 ndash 413 6 8 13 14

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 11

Section 72 ndash Developing Quadratic FunctionsEquations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadratic function f(x) = ax2 + bx + c

Example Determine the zeros for f(x) = x2 ndash 4x ndash 12

Determining Roots of a Quadratic Equation ax2 + bx + c = 0

Example Determine the roots for 5x2 + 7x ndash 6 = 0

Procedure set y = 0 OR f(x) = 0

solve ax2 + bx + c = 0 by factoring

Procedure arrange the quadratic equation to the form ax2 + bx + c = 0

solve ax2 + bx + c = 0 by factoringquadratic formula or isolating the

variable

Goal

Developing Quadratic FunctionsEquations

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 12

(II) Developing a quadratic equationfunction given rootszeros

Example Determine the quadratic function y = ax2 + bx + c

that has zeros ndash2 and 6

Example Determine the quadratic equation ax2 + bx +c = 0

that has roots ndash2 and 3

5

How do we reverse the procedure for attaining zeros

so we can develop the function

(i)

(ii)

(iii)

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 13

Example Determine the quadratic equation that has roots

(a) 3

4 119886119899119889 minus

2

3 (b) plusmn2radic5 (c) 3 plusmn 4radic2

How will this be evaluated Potential multiple choice questions

1 Which quadratic function has zeros 2 1___

(A) 2 4y x

(B) 2 2y x

(C) 2 2y x

(D) 2 2y x

2 Which equation has roots 3

2 and 4

2___

(A) 24 11 6 0x x

(B) 24 5 6 0x x

(C) 24 5 6 0x x

(D) 24 11 6 0x x

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 14

Practice Problems

1 Given the zeros determine the quadratic function

(a) minus3

5 119886119899119889

1

2 (b) 1 plusmn 2radic3

(c) 0 119886119899119889 4

3 (d) plusmn4radic7

2 Given the roots determine the quadratic equation

(a) 2

7 119886119899119889 5 (b) plusmnradic3

Answers

1(a) y = 10x2 + x ndash 3 (b) y = x2 ndash 2x ndash 11 (c) y = 3x2 ndash 4x (d) y = x2 ndash 112

2(a) y = 7x2 ndash 37x + 10 (b) y = x2 ndash 3

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 15

x- 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2

y

- 6

- 5

- 4

- 3

- 2

- 1

1

2

Quiz Review Quadratic Equations

1 Factor the following

(a) x2 + 5x (b) 9x2 ndash 1 (c) x2 + 6x + 8 (d) 3x2 ndash 7x + 2 (e) 5x2 + 15x ndash 20

2 Determine the xndashintercepts of the following graphs

(a) (b)

3 A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation for the

height of the missile at 160 m

(b) At what times does the missile reach a

height of 160 m

(c) Write the quadratic equation for the

height of the missile at 100 m

(d) At what times does the missile reach a

height of 100 m

x-2 2 4 6 8

y

-4

-2

2

4

6

8

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 16

4 Use factoring to determine the zeros of the following quadratic functions

(a) y = x2 ndash 3x (b) y = 2x2 + 8x (c) y = x2 ndash 16 (d) y = 4x2 ndash 9

(e) y = x2 ndash 3x ndash 10 (f) y = x2 ndash 10x + 24 (g) y = 2x2 + 9x + 10

(h) y = 6x2 ndash 5x ndash 6 (i) y = 2x2 ndash 50 = 0 (j) y = 3x2 + 3x ndash 36

5 Use factoring to determine the roots of the following equations

(a) x( x + 4 ) = 12 (b) 2( x2 + 1 ) = 5x (c) 025x2 = 05x + 075

(d) ndash14x2 + 84x ndash 7 = 0 (e) 42x2 + 63x = ndash21 (f) 2

1x2 = 18

(g) 3

4x2 + x ndash

3

1

= 0 (h)

4

1x2 + 2x = ndash4

6 The path of a missile shot into the air from a ship is modeled by the

quadratic function h(t) = ndash 49t2 + 294t + 784 where h represents the

height in meters and t is time in seconds

(a) Determine how long it takes for the missile to hit the ocean

(b) Determine at what times the missile reaches a height of 1029 m

7 An osprey dives toward the water to catch a salmon Its height above the water

in metres t seconds after it begins its dive is approximated by h(t) = 5t2 ndash 25t + 45

Algebraically determine the time it takes for the osprey to catch the salmon and then

reach a return height of 25 m

8 Travel World Agency has 25 people signed up for a vacation The revenue for the

vacation is modeled by the function R(p) = ndash 50p2+400p + 2500 where R(p) represents

the revenue and p represents the number of additional people to sign up How many

additional people must sign up for the revenue to reach $1500

9 Determine the quadratic function y = ax2 + bx + c that has the given zeros of a function

(a) ndash 4 and 2 (b) 0 and 3

2

(c) 23

(d) 3 and

5

1

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 17

10 Determine the quadratic equation ax2 + bx + c = 0 that has the given roots

(a) ndash 3 and 1 (b) 2

1 and

4

3

(c) 3

(d) ndash 2 and

2

3

11 Which function has zeros 2 12 Which equation has roots ndash 3 and 3

2

(A) 2 4y x (A) 3x2 ndash 7x ndash 6 = 0

(B) 2 2y x (B) 3x2 + 7x ndash 6 = 0

(C) 2 2y x (C) 3x2 + 7x + 6 = 0

(D) 2 2y x (D) 3x2 ndash 7x + 6 = 0

Answers

2(a) x = ndash 5 ndash 1 (b) x = 2 6

3(a) 160 = ndash 20t2 + 120t (b) 2 sec 4 sec (b) 100 = ndash 20t2 + 120t (d) 1 sec 5 sec

4(a) x = 0 3 (b) x = ndash 4 0 (c) x = ndash 4 4 (d) x = 2

3

2

3 (e) x = ndash 2 5

(f) x = 4 6 (g) x = 2

5 ndash 2 (h) x =

3

2

2

3 (i) x = ndash 5 5 (j) x = ndash 4 3

5(a) x = ndash 6 2 (b) x = 2

1 2 (c) x = ndash 1 3 (d) x = 1 5 (e) x = ndash 1

2

1

(f) x = ndash 6 6 (g) x = ndash 1 4

1 (h) x = ndash 4

6(a) t = 8 sec (b) t = 1 sec 5 sec 7 t = 4 sec 8 10 people

9(a) y = x2 + 2x ndash 8 (b) y = 3x2 ndash 2x (c) y = x2 ndash 18 (d) y = 5x2 ndash 14x ndash 3

10(a) x2 + 2x ndash 3 = 0 (b) 8x2 ndash 2x ndash 3 = 0 (c) x2 ndash 3 = 0 (d) 2x2 + x ndash 6 = 0

11 D 12 B

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 18

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(I) Applying the quadratic formula to solve a quadratic equation

Example

A missile fired from ground level is modeled by the quadratic function

h(t) = ndash20t2 + 120t where h(t) represents height in meters and t is

the time in seconds

(a) Write the quadratic equation that could be used to determine the

times where the missile attains a height of 80 m

(b) Express the quadratic equation from (a) above in standard form

(c) Can we solve the quadratic equation from (b) by factoring

t1 2 3 4 5 6 7 8

h(t)

20406080

100120140160180200

h(t) = ndash20t2 + 120t

Goal

Applying the Quadratic Formula to Determine the Roots of a

Quadratic Equation

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 19

Apply the quadratic formula to solve the quadratic equation to determine the times

when the missile attains a height of 80m

Solving Quadratic Equations by Quadratic Formula

We can solve quadratic equations in the form ax2 + bx + c = 0 that do not

factor by applying the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 20

(II) Determining EXACT ROOTS of a Quadratic Equation by

applying Quadratic Formula

What is an EXACT ROOT

Example Use a calculator to approximate radic2

Example Simplify the EXACT ROOT

(a) radic80 (b) radic27

Example Determine the EXACT ROOTS for

(a) x2 ndash 2x = 17 (b) 4x(x ndash 3) = ndash7

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 21

(III) Solving Quadratic Equations that factor by applying the

Quadratic Formula

Example

Solve by factoring 3x(x + 2) = ndash4(x ndash 2)

Example

Solve by applying the quadratic formula 3x(x + 2) = ndash4(x ndash 2)

P419 ndash 420 2a d 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 22

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 2)

(I) Investigating the type of attainable roots by application of the

quadratic formula

Solve each equation by applying the quadratic formula

(A) x2 + 2x ndash 8 = 0 (B) x2 + x ndash 4 = 0

Investigating Attainable Roots

(i) What is the value under the square root in (A) Is it a perfect square

How many roots exist and are they exact or approximate

Goal

Applying the Quadratic Formula to Solve Quadratic

Equations and Distinguishing the type of Attainable Roots

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 23

Investigating Attainable Roots

(ii) What is the value under the square root in (B) Is it a perfect square

How many roots exist and are they exact or approximate

(iii) What values of b2 ndash 4ac could lead to approximate answers

(iv) What values of b2 ndash 4ac could lead to exact answers

(v) Which equation could also be solved by factoring

(vi) What connection can be made between an equation that is factorable

and the value of b2 ndash 4ac

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 24

(II) Investigating the number of zero(s)root(s) of a

quadratic functionequation

A quadratic functionequation can have two one or no real zerosroots

pending on the value attained underneath the root of the quadratic formula

119909 =minus119887plusmnradic1198872minus4119886119888

2119886

The result of b2 ndash 4ac predicts the number of zeros (x-intercepts)roots

Investigate

In each case determine how the value of b2 ndash 4ac compares to 0 (gt lt =)

to represent what has occurred graphically

(a) Calculate the value of b2 ndash 4ac and how it compares

to 0 (gt lt =) for y = x2 ndash 5x + 4 and sketch the graph

(Check out httpswwwdesmoscom)

The value of b2 ndash 4ac and

how it compares to 0 can

be used to predict whatrsquos

happening graphically

D = b2 ndash 4ac

x

y

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 25

(b) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 + 4x + 4 and sketch the graph (Check out httpswwwdesmoscom)

(c) Calculate the value of b2 ndash 4ac and how it compares to 0 (gt lt =) for

y = x2 ndash 4x + 5 and sketch the graph (Check out httpswwwdesmoscom)

When b2 ndash 4ac 0

it means that _____ zero(s)root(s) are attained

x

y

When b2 ndash 4ac 0

it means that ___________ zero(s)root(s) are attained

x

y

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 26

(III) Investigating the type of attainable root(s) from a quadratic equation

and making a connection to the corresponding graph of the quadratic

function

Example

Apply the quadratic formula to determine the roots of the

quadratic equation ndash2x(x ndash 7) = 5(2x + 1)

Express the quadratic equation above (ax2 + bx + c = 0) as a quadratic function

(y = ax2 + bx + c) and sketch the graph

(Check out httpswwwdesmoscom)

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ solution for

the quadratic equation

x

y

If the value of the radicand (or b2 ndash 4ac) is

_____________ in the quadratic formula

then there is _____________ zeros for the

quadratic function

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 27

Practice Problems

Determine the exact roots for

1 2x2 = 8x ndash 5 2 2x2 ndash 2x ndash 3= 0

3 x(x ndash 1) ndash 3 = 5x 4 ndash3x2 ndash 2x = 2

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 28

5 1

31199092 + 4119909 = minus15

Answers

1 4 plusmn radic6

2 2

1 plusmn radic7

2 3 3 plusmn 2radic3

4 minus2 plusmn radicminus20

6 5

minus12 plusmn radicminus36

2

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 29

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 3)

(I) Modeling Projectile Motion

Example 1 Quadratic Equation is given that models projectile motion

in seconds

How long does it take for the football to attain a height of 15 feet

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 30

(II) Modeling Revenue

Example

A concert promoterrsquos profit P(s) in dollars can be modeled by the function

P(s) = ndash8s2 + 950s ndash 250

where s is the price of a ticket in dollars

(a) If the promoter wants to earn a profit of $20 000 what should be the

price of the ticket

(b) Is it possible for the promoter to earn a profit of $30 000 Explain

P420 ndash 421 7 8a c 10

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 31

QUADRATIC EQUATIONS

SAMPLE INCLASS ASSIGNMENT

QUADRATIC FORMULA

PART A MULTIPLE CHOICE ( Value 8 )

Place the letter of the correct response in the space provided to the right

1 Which quadratic function below has xndashintercepts of ndash4 and 1 and a yndashintercept of 4 1

(A) (B)

(C) (D)

2 Which are the zeros of the quadratic function f(x) = ( 3x ndash 4 )( x + 2 ) 2

(A) x = 2 x = 3

4 (B) x = ndash 2 x =

3

4

(C) x = ndash2 x =

3

4 (D) x = 2 x =

3

4

x- 2 - 1 1 2 3 4 5

y

- 6- 5- 4- 3- 2- 1

123456

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 6- 5- 4- 3- 2- 1

123456

x- 2 - 1 1 2 3 4 5

y

- 3

- 2

- 1

1

2

3

4

5

6

7

x- 5 - 4 - 3 - 2 - 1 1 2

y

- 3

- 2

- 1

1

2

3

4

5

6

7

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 32

3 Which quadratic equation has roots of 0 and 3

2 3

(A) 3x2 ndash 2x = 0 (B) 2x2 ndash 3x = 0 (C) 3x2 + 2x = 0 (D) 2x2 + 3x = 0

4 What are the roots of the quadratic equation 9 ndash 4x2 = 0 4

(A) x = 2

3

(B) x =

3

2

(C) x =

4

9

(D) x =

9

4

5 A golf ball is struck and its trajectory is modeled by the function h(t) = ndash4t2 + 20t

where h(t) represents height in meters and t is time in seconds Determine the time it

takes for the golf ball to hit the ground after it has been struck 5____

(A) 0 sec (B) 25 sec (C) 4 sec (D) 5 sec

6 Which graph represents a quadratic function with no real zeros 6

(A) (B)

(C) (D)

function 7____

(A) There are three xndashintercepts (B) There are two xndashintercepts

(C) There is one xndashintercept (D) There are zero xndashintercepts

x

y y

x

y

x

y

x

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 33

8 Mark used the quadratic formula as shown below to solve the quadratic equation x2 + 7x ndash 1 = 0

He made an error in his calculations In which step did Mark first make his mistake 8____

(A) Step 1 (B) Step 2 (C) Step 3 (D) Step 4

Step 1 )1(2

)1)(1(477 2

Step 2 2

4497

Step 3 2

457

Step 4 2

537

PART B QUESTIONS ( Value 16 )

Answer each question in the space provided Show all your workings to ensure full marks

9 Solve by factoring ( 4 )

x( x + 4 ) = 12

10 Determine the quadratic equation that has roots of 2

1

and

3

2 ( 4 )

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 34

11 Determine the EXACT ROOTS for the quadratic equation 2x(x ndash 3) = 2(x + 4) ( 4 )

h represents height in meters and t is time in seconds Determine the time(s) when the missile

reaches a height of 100 m ( 4 )

ANSWERS

1 D 2 C 3 B 4 A 5 D 6 D 7 B 8 B

9 x = ndash6 and x = 2 10 6x2 ndash x ndash 2 = 0 11 119909 = minus2 plusmn 2radic2

12 4 sec and 6 sec

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 35

x

25 ft

x

x x

x x

12 ft

Section 73 ndash Solving Quadratic Equations by Quadratic Formula

(Day 4)

REMEMBER Area = length x width

(III) Quadratic Equations Developed through Area Problems

surrounded by a paving stone border of uniform width on three sides The

dimensions of the driveway including the paving stone border is 25 feet by

12 feet The pavement in the driveway represents 240 ft2 Determine the

quadratic equation that models the area of the driveway and use it to determine

the width of the paving stone border

Driveway

Goal

Modeling a Problem with a Quadratic Equation and Solving

the Equation

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 36

Playg

roun

d

50m

30m

Safety Zone

Example 2 Modeling Area Expansion Problems

A school playground is rectangular and has a length of 50 m and a width of 30 m

as shown A safety zone of uniform width surrounds the playground If the entire

area is 3500m2 what is the width of the safety zone

Example 3

A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform

width The combined area of the lawn and flower bed is 165 m2 What is the width of

the flower bed

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 37

x x

x

x

Practice Problems

1 A strip of pavement of the same width will be constructed around three sides

of 30 m by 20 m Warehouse building The total area of the building and

pavement is 1500 m2 Develop the quadratic equation that models the area

of the Warehouse and pavement and use it to determine the width of the

pavement strip

2 A rectangular picture 7 cm by 5 cm is surrounded by a uniform wooden

frame so that the total area of frame and picture is 143 cm2 Set up an

equation to model this situation and use it to algebraically determine the

width of the wooden frame

Warehouse

x

x x

x

x x

30 m

20 m

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 38

3 A rectangular garden measuring 15 m by 20 m has a uniform strip removed from

the edge of one length and the edge of one width to make a concrete walkway

The area of the remaining garden is 200 m2 Develop a quadratic equation that

models the area of the new garden and solve it to determine the

width of the concrete walkway

4 A carpenter will renovate a 10 ft by 12 ft rectangular room by knocking

down and moving two walls A uniform extension out of one length and

width wall will double the existing area Develop the quadratic equation

that models the area of the new room and use it to determine extension of

each wall

20 m

15 m

x

x

Answers 1 10 m 2 3 cm 3 315 m 4 453 ft

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 39

Section 74 ndash Applications of Quadratic Equations

(I) Modeling Projectile Motion based on a verbal description

Example Developing a quadratic function that models projectile motion

A baseball is thrown from an initial height of 3 m and reaches a maximum

height of 8 m 2 seconds after it is thrown What time does the ball hit the

ground

Height h(t)

time t

Goal

Solve Problems that Involve Quadratic Equations

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 40

(II) Modeling Projectile Motion based on a given function

Example

A diverrsquos path when diving off a platform is given by d = ndash5t2 + 10t + 20

where d is the distance above water in feet and t is the time in seconds

(a) How high is the diving platform

(b) After how many seconds is the diver 25 feet above the water

(c) When does the diver enter the water

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 41

(III) Applying Quadratic Equations to Numeracy Problems

Consecutive means one number after another so if the first number is x the next

number will be x+1or x ndash 1

Consecutive even (or odd) numbers you will have x and x + 2 or x ndash 2

You cannot have a negative length or width

Example

Determine two consecutive positive even numbers that have a product of 48

Example

The sums of squares of two consecutive even integers is 100 Determine the

integers

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 42

PRACTICE PROBLEMS

1 A missile is shot into the air from a ship The height of the missile

above sea level in metres t seconds after being shot is approximated

by h(t) = ndash6t2 + 26t + 4

Algebraically determine

(a) the times when the missile attains a height of 12 m

(b) the time when the missile hits the ocean

t

h(t)

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 43

2 An owl perched in a tree ascends to a maximum height of 30 metres at

4 seconds It spots a rat on the ground and descends to strike it at 10

seconds

(a) Determine the quadratic function that describes the flight path

(b) Determine the time(s) it takes for the owl to attain a height of 20 m

t

h

P430 ndash P431 3 5 7 8

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 44

Chapter 7 Quadratic Equations Test Review

QUADRATIC FORMULA a

acbbx

2

42

1 Solve the following quadratic equations using factoring

a) 61199092 = 3119909 b) 41199092 minus 9 = 0 c) 31199092 = 119909 + 10 d) 2119909(119909 + 4) = 10

e) 119909(119909 + 10) = 4119909 + 16 f) minus251199092 = 80 minus 30119909 g) 1

41199092 + 2119909 minus 5 = 0

2 Solve the following using the quadratic formula

a) 1199092 minus 4119909 + 1 = 0 b) 1199092 + 10119909 minus 2 = 0

3 Determine the EXACT roots of the following

d) 31199092 + 11119909 = 4 b) e) 211199092 = 189

a) 119909 = 9 119909 = 2 b) 119909 = minus3

4 119909 = 1 c) 119909 = minus

1

4 119909 =

3

5 d) 119909 = plusmnradic5

5 A rocket is fired into the air according to the equation ℎ(119905) = minus21199052 + 4119905 + 48

where 119905 is the time in minutes and ℎ is the height in meters

(a) Determine the time(s) the rocket is at a height of 32 meters

(b) Determine when the rocket hits the ground

6 A missilersquos path when fired from a ship is given by h(t) = ndash3t2 + 2t + 8

where h(t) is the height of the missile in metres and t is the time in seconds

(a) When does the missile hit the water

(b) Approximately when does the missile reach a height of 4m

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 45

7 The revenue made by a drama theater is represented by R = ndash20x2 + 80x + 3200

where x is the number of shows the drama group performs How many shows

should the drama group have to make a profit of $ 2000

8 A rectangular swimming pool has length 30 m and width 20 m There is a deck of

uniform width surrounding the pool The area of the pool is the same as the area of the

deck Write a quadratic equation that models this situation and use it to determine the

width of the deck

9 A driveway has dimensions of 8m x 18m Then a flower bed of uniform width

is added to two sides of the driveway What is the width of the flower bed if the

total combined area is 200 1198982

10 Susan decides to build a uniform deck around her pool which has dimensions of

20 m by 10 m If the total area of the pool and deck measures 300m2 then write a

quadratic equation that models this situation and use it to determine the width of

the uniform strip denoted by x

Pool

x

x

x

x

x

x

x

x

30 m

20 m

Driveway

x x

x

x

20 m

10 m

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 46

11 A rectangular garden measuring 20 m by 15 m has a uniform strip removed

from the edge of one length and the edge of one width to make a concrete

walkway If the area of the remaining garden is 204 m2 what will be the

width of the concrete walkway

12(a) Which graph represents a quadratic function with two unequal real zeros

(b) Which graph represents a quadratic function with two equal real zeros

(c) Which graph represents a quadratic function with two unequal unreal zeros

(A) (B)

(C) (D)

13 Determine the number of roots and type of roots for each of the following using b2 ndash 4ac

a) 3x2 ndash 2x + 1 = 0 b) x2 + 6x + 9 = 0 c) 2x2 + 5x ndash 3 = 0

14 The product of two odd consecutive integers is 63 Determine the integers

20 m

15 m

x

x

x

y y

x

y

x

y

x

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6

Unit III Quadratic Equations 47

15 The sum of the squares of two even consecutive integers is 52 Determine the integers

ANSWERS

1(a) 2

10x (b)

2

3x (c) 2

3

5x (d) 15x

(e) 28x

(f) 84x

(g) 210x

2(a) 32 x (b) 335 x 3(a) 51x (b) 31x (c) 23x

(d)

3

14x

3(e) 3x 4(a) 018112 xx (b) 034 2 xx (c) 03720 2 xx (d) 052 x

5(a) 4 sec (b) 6 sec 6(a) 2 sec (b) 154 sec 7 10 shows 8 06001004 2 xx 5 m

9 width = 2 m 10 0100604 2 xx 151 m 11 width = 3 m 12(a) B (b) C (c) D

13(a) 0 roots (b) 1 root (c) 2 roots 14 ndash 9 ndash 7 7 9 15 ndash 6 ndash 4 4 6