17

Section 1.4 - Quadratic Equations

Basic Complex Number

2 1i 1i 1 i

The number i is called the imaginary unit.

Example

8 2 2i 2 2i

7 7 7 7i i

22 7i

7

Complex number is written in a form: z a bi

a is the real part

b is the imaginary part

Conjugate of a complex number is a bi a bi

A quadratic equation in x is an equation that can be written in the general form:

2 0x xba c where a, b, and c are real numbers,

2 24 03x x 4 23 ba c

Solving Quadratic Equations by Factoring

The Zero-Product Principle

If AB = 0 then A = 0 or B = 0.

Example

Solve 26 7 3 0x x

Solution

(3 1)(2 3) 0x x

3 1 0x 2 3 0x

3 1x 2 3x

13

x 32

x

18

The Square Root Property

If u is an algebraic expression and d is a nonzero real number, then u2 = d has exactly two solutions:

2 , If u d then u d or u d Equivalently,

2 If u d u d .

Example

a) 3x2 – 21 = 0

23 21x

2 7x 7 x

b) 5x2 + 45 = 0

25 45 x

2 9 x

9 x x = 3i

c) (x + 5)2 = 11

x + 5 = 11 115 x

Completing the Square

If 2x bx is a binomial, then by adding 2

2b

which is the square of half the coefficient of x. a perfect

square trinomial will result. That is.

2 2

22 2

bx b bx x

2 2

22

12

bx bx xb

Example

Solve: 2 4 1 0x x

2 4 1x x

2 2

4 42 2

2 4 1x x

22 4 2 1 4x x

22 5x

x + 2 = 5

2 5x

19

Quadratic Formula

(Using Completing the Square)

2 0ax bx c

2ax bx c

2 b cx x

a a

2

22

1 12 2

b cxa

b ba

xaa

2 2

22 4

b c bxa a a

2

24

b caa

2

24

4

b ac

a

2

2

42

4

b acbxa

a

2 42 2

b acbxa a

2 42

b b acx

a

*** 2

2 0 42

cxbxa x ba

cb a

2 4 c 0 2

2 4 0 2

2 4 0 O ( )

b a Real numbers

b ac Complex numbers

b ac ne solution repeated

20

Example

Solve: 22 2 1 0x x

Solution

22 1a b c

2 4

2x

cb b a

a

22 4( )( )

2(

2

2)

2 1

= 4

842

2 12

4

122

4 4

2 2 3

4

2 31

2 4

2 1 3

4

312 2

1 3

2

3 31 1,

2 22 2

312 2

Example

Solve 2 4 2x x

Solution

2 4 2 0x x

41 2a b c

2 4

2x

cb b a

a

( 4) ( 4)2 4(1)

(1

2)

)2

(

4 16 8

2

4 8

2

4 2 2

2

21

2

2

2 2

2 2

The solution set is 2 2

Example

Solve: 2 2 2 0x x

Solution

21 2a b c

2( ) ( ) 4( )( )

2

2

(

2

1

21

)

x

2 4 8

2

2 4

2

42

2 2

2 2

2

i

2

12

i

2

2

(1 )i

1 i

1 i

Pythagorean Theorem

The sum of the squares of the lengths of the legs of a right triangle equals

the square of the length of the hypotenuse. If the legs have lengths a and

b, and the hypotenuse has length c, then:

2 2 2a b c

Example

A ladder that is 17 feet long is 8 feet from the base of a wall. How far up the wall does the ladder reach?

Solution

2 2 28 17y

2 2 217 8y

2 217 158y ft

17 ft

8 ft

y

22

Height of a Projected Object (Position Function)

An object that is falling or vertically projected into the air has its height above the ground, s(t), in feet,

given by

0 0

2( ) 16s t t v t s

0v is the original velocity (initial velocity) of the object, in feet per second

t is the time that the object is in motion, in second

0s is the original height (initial height) of the object, in feet

Example

If a projectile is shot vertically upward from the ground with an initial velocity of 100 ft per sec,

neglecting air resistance, its height s (in feet) above the ground t seconds after projection is given by

216 100s t t

a) After how many seconds will it be 50 ft above the ground?

b) How long will it take for the projectile to return to the ground?

Solution

a) After how many seconds will it be 50 ft above the ground?

2 10 16 005 t t

216 100 50 0t t

28 50 25 0t t

2( 50) ( 50) 4(8)(25)

2(8)t

50 1700

16

50 1700

160.55t

50 1700

165.70t

b) How long will it take for the projectile to return to the ground?

20 16 100t t

0 4 (4 25)t t

4 0t

4 25 0t

0t

4 25t

254

6.25t

23

Exercises Section 1.4 - Quadratic Equations

1. Solve: 2 25x

2. Solve: 25 45 0x

3. Solve: 24 12x

4. Solve: 24 1 20x

5. Solve 2 6 7x x

6. Solve 26 3 2x x

7. Solve: 23 2 7x x

8. Solve: 23 6 10x x

9. Solve: xx 25 2

10. Solve: 25 2 3x x

11. Solve: 2 8 15 0x x

12. Solve: 2 5 2 0x x

13. Solve: 2 6 10 0x x

14. Solve: 22 3 4 0x x

15. Solve: 2 8 0x x

16. Solve: 22 13 1x x

17. Solve: 2 3 3 0r r

18. Solve: 3 8 0x

19. Solve for the specified variable 2

4, dA for d

20. Solve for the specified variable 2 0 ( 0), rt st k r for t

21. A rectangular park is 6 miles long and 2 miles wide. How long is a pedestrian route that runs

diagonally across the park?

22. What is the width of a 25-inch television set whose height is 15 inches?

24

23. A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by

12 meters. A path of uniform width is to surround the garden. If the area of the garden and path

combined is 378 square meters, find the width of the path.

24. A pool measuring 10 m by 20 m is surrounded by a path of uniform width. If the area of the pool

and the path combined is 600 m2, what is the width of the path?

25. A boat is being pulled into a dock with a rope attached to the boat at water level. Where the boat is

12 ft from the dock, the length of the rope from the boat to the dock is 3 ft longer than twice the

height of the dock above the water. Find the height of the dock.

26. Logan and Cassidy leave a campsite, Logan biking due north and Cassidy biking due east. Logan

bikes 7 km/h slower than Cassidy. After 4 hr, they are 68 km apart. Find the speed of each bicyclist.

27. Towers are 1482 ft. tall. How long would it take an object dropped from the top to reach the ground?

Given s = 16t2

25

28. The formula P = 0.01A2+ 0.05A + 107 models a woman's normal Point systolic blood pressure, P,

an age A. Use this formula to find the age, to the nearest year, of a woman whose normal systolic

blood pressure is 115 mm Hg.

29. A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut

from the four corners, and the flaps folded upward to form an open box. If the volume of the box is

832 3in , what were the original dimensions of the piece of metal?

30. An astronaut on the moon throws a baseball upward. The astronaut is 6 ft., 6 in., tall, and the initial

velocity of the ball is 30 ft. per sec. The height s of the ball in feet is given by the equation

22.7 30 6.5t ts

Where t is the number of seconds after the ball was thrown.

a) After how many seconds is the ball 12 ft above the moon’s surface?

b) How many seconds will it take for the ball to return to the surface?

31. The bar graph shows of SUVs (sport utility vehicles0 in the US, in millions. The quadratic equation

2.00579 .2579 .9703S x x models sales of SUVs from 1992 to 2003, where S represents sales in

millions, and x = 0 represents 1992, x = 1 represents 1993 and so on.

a) Use the model to determine sales in 2002 and 2003. Compare the results to the actual figures of 4.2

million and 4.4 million from the graph.

b) According to the model, in what year do sales reach 3.5 million? Is the result accurate?

32. Cynthia wants to buy a rug for a room that is 20 ft wide and 27 ft long. She wants to leave a uniform

strip of floor around the rug. She can afford to buy 170 square feet of carpeting. What dimension

should the rug have?

26

33. Erik finds a piece of property in the shape of a right triangle. He finds that the longer leg is 20 m

longer than twice the length of the shorter leg. The hypotenuse is 10 m longer than the length of the

longer leg. Find the lengths of the sides of the triangular lot.

34. An open box is made from a 10-cm by 20-cm piece of tin by cutting a square from each corner and

folding up the edges. The area of the resulting base is 96 2cm . What is the length of the sides of the

squares?

23

Solution Section 1.4 - Quadratic Equations

Exercise

Solve: 2 25x

Solution

25x

5i

Exercise

Solve: 24 12x

Solution

4 12x

4 12x 12 4(3) 2 3

4 2 3x

Exercise

Solve: 25 45 0x

Solution

5x2 = 45

545x

x2 = 9

x = 3

Exercise

Solve: 24 1 20x

Solution

4 1 20x

4 1 2 5x

1 2 54

x

24

Exercise

Solve 2 6 7x x

Solution

)1(2

)7)(1(4)6()6( 2 x

2

86

2

226

2

232

23

Exercise

Solve 26 3 2x x

Solution

0236 2 xx

0236 2 xx

)6(2

)2)(6(433 2 x

12

393

39312 12

i

3914 12

i

Exercise

Solve: 23 2 7x x

Solution

2 a3 2 7 0 = , , = 3 b 2x x c = - 7

2 4

2

b b ax

a

c

2 4(3(2 )() (2

(3)

))

2

25

2 886

2 4(22)

6

2 2 226

2 1 22

6

1 223

2213 3

x

Exercise

xx 1063 2

Solution

23 10 6 0x x

2 4

2

b b ax

a

c

2( 10) ( 10)

32

3)

( )

4(

(6)

10 100 72

6

10 28

6 6

5 2 7

3 6

5 7

3 3

26

Exercise

xx 25 2

Solution

025 2 xx

)5(2

)2)(5(42)1()1( x

10

4011

10

391

10

391 i

39110 10

i

Exercise

Solve: 25 2 3x x

Solution

25 2 3 0x x

2( 2) ( 2) 4( )(3)

2( )

5

5x

2 4 60

10

2 56

10

2 4(14)

10

i

2 2 14

10

i

2 14210 10

i

1415 5

i

27

Exercise

Solve: 2 8 15 0x x

Solution

12 4(8 )(15)

2

8

1( )x

8 64 60

2

8 4

2

8 2

2

8 2 6 322

8 2 10 522

Exercise

Solve: 2 5 2 0x x

Solution

2 4( )5 (5 2)

2

1

(1)x

5 25 8

2

5 17

2

5 17

2 2

28

Exercise

Solve: 2 6 10 0x x

Solution

2 6 10x x

2

22

6 6102 2

6x x

22 2(3) 3 10 9x x

2( 3) 19x

3 19x

3 19x

Exercise

Solve: 22 3 4 0x x

Solution

22 3 4x x

2 32

2x x

2 2

3 31 12 2

2 322 2

2x x

22 3 3 9

2 4 162x x

2

3 414 16

x

3 414 16

x

4134 4

x

29

Exercise

Solve 2 8 0x x

Solution

2 4

2

b b acx

a

21 1 4 1 8

2 1

1 1 32

2

1 31

2

1 31

2

i

Exercise

Solve 22 13 1x x

Solution

22 13 1 0x x

2 4

2

b b acx

a

213 13 4 2 1

2 2

13 169 8

4

13 177

4

Exercise

Solve 2 3 3 0r r

Solution

23 3 4 1 3

2 1r

30

3 9 12

2

3 21

2

Exercise

Solve: 3 8 0x

Solution

2( 2) 2 4 0x x x 3 3 2 2( )a b a b a ab b

2 0x 2 2 4 0x x

2x 2

( 2) ( 2) 4(1)(4)

2(1)x

2 12

2

2 2 3

2

i

2

2

1 3i

1 3i

The solution set is 2, 1 3i

Exercise

Solve for the specified variable 2

4, dA for d

Solution

24 4

4dA

2

44 4 dA

24A d

24A d

2 4Ad

24A d

4Ad

2A

31

2A

2 A

Exercise

Solve for the specified variable 2 0 ( 0), rt st k r for t

Solution

2 4

2

cb ab

at

2( ) ( ) 4( )( )

2( )

s s r k

rt

24

2

s s rk

rt

Exercise

A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12

meters. A path of uniform width is to surround the garden. If the area of the garden and path combined is

378 square meters, find the width of the path.

Solution

(15 2 )(12 2 )Area x x

378 (15 2 )(12 2 )x x

2378 180 30 24 4x x x

20 180 54 4 378x x

20 4 54 198x x

24 54 198 0x x

2(54) (54) 4(4)( 198)

2(4)x

54 6084

8

54 78

16.

54 78 8

54 7888

5

3 x

x

32

Exercise

A rectangular park is 6 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally

across the park?

Solution

2 2 26 2d

2 40d

40 6.32 d miles

Exercise

A pool measuring 10 m by 20 m is surrounded by a path of uniform width. If the area of the pool and the

path combined is 600 m2, what is the width of the path?

Solution

A lw

600 (20 2 )(10 2 )x x

2600 200 40 20 4x x x

20 600 200 60 4x x

20 400 60 4x x

20 100 15x x

215 100 0x x

215 15 4(1)( 100)

2(1)x

15 6252

15 252

15 25 5

20

2

The width of the path is 5 m.

33

Exercise

A boat is being pulled into a dock with a rope attached to the boat at water level. Where the boat is 12 ft

from the dock, the length of the rope from the boat to the dock is 3 ft longer than twice the height of the

dock above the water. Find the height of the dock.

Solution

2 2 2(2 3) 12h h

2 24 12 9 144h h h

2 24 12 9 144 0h h h

23 12 135 0h h

2 4 45 0h h

( 9)( 5) 0h h

, 59 h

Height = 5 ft.

Exercise

What is the width of a 25-inch television set whose height is 15 inches?

Solution

2 2 215 25w

2 2 225 15w

2 225 15w

20 w in

34

Exercise

Logan and Cassidy leave a campsite, Logan biking due north and Cassidy biking due east. Logan bikes 7

km/h slower than Cassidy. After 4 hr, they are 68 km apart. Find the speed of each bicyclist.

Solution

22 24 4( 7) 68r r

2 216 16( 14 49) 4624r r r

2 216 16 224 784 4624r r r

232 224 784 4624 0r r

232 224 3840 0r r

2 7 120 0r r

8, 15r

' 15 /Cassidy s km h

' 8 /Logan s km h

Exercise

Towers are 1482 ft tall. How long would it take an object dropped from the top to reach the ground?

Given s = 16t2

Solution

1482 = 16t2

21482

16t

148216

t

t 9.624 sec

35

Exercise

The formula P = 0.01A2+ 0.05A + 107 models a woman's normal Point systolic blood pressure, P, an age

A. Use this formula to find the age, to the nearest year, of a woman whose normal systolic blood pressure

is 115 mm Hg.

Solution

0.01A2

+ 0.05A + 107 = 115

0.01A2

+ 0.05A – 8 = 0

A = 2.05 .05 4(.01)( 8)

2(.01)

.05 .0025 .32

.02

.05 .567

.02

2689.2502.

567.05.

) a (31 02.

567.05.SolutionNot

Exercise

A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the

four corners, and the flaps folded upward to form an open box. If the volume of the box is 832 3in , what

were the original dimensions of the piece of metal?

Solution

10 wl

Bottom width: 4w

Bottom length: 64104 wwl

2)4)(6( wwlwhV

)2464(2 2 www

4842 2 ww

22 4 48 832w w

22 4 880 0w w

2 2 440 0w w

( 22)( 20) 0w w

22 0w 20 0w

22w 20w

Width of the metal is 20 in by the length (20+10) 30 in.

36

Exercise

An astronaut on the moon throws a baseball upward. The astronaut is 6 ft., 6 in., tall, and the initial

velocity of the ball is 30 ft. per sec. The height s of the ball in feet is given by the equation

22.7 30 6.5t ts

Where t is the number of seconds after the ball was thrown.

a) After how many seconds is the ball 12 ft above the moon’s surface?

b) How many seconds will it take for the ball to return to the surface?

Solution

a) After how many seconds is the ball 12 ft above the moon’s surface?

21 2.7 30 6.2 5t t

20 2.7 30 6.5 12t t

20 2.7 30 5.5t t

230 (30) 4( 2.7)( 5.5) 30 29

2( 2.7) 5.4t

30 295.4

t

30 295.4

t

10.9sect

.12sect

b) How many seconds will it take for the ball to return to the surface?

22.7 30 6.0 5t t

230 (30) 4( 2.7)(6.5) 30 31.15

2( 2.7) 5.4t

30 31.155.4

t

30 31.155.4

t

11.32t

0.212t

It will take 11.32 sec.

37

Exercise

The bar graph shows of SUVs (sport utility vehicles0 in the US, in millions. The quadratic equation

2.00579 .2579 .9703S x x models sales of SUVs from 1992 to 2003, where S represents sales in millions,

and x = 0 represents 1992, x = 1 represents 1993 and so on.

a) Use the model to determine sales in 2002 and 2003. Compare the results to the actual figures of 4.2 million

and 4.4 million from the graph.

b) According to the model, in what year do sales reach 3.5 million? Is the result accurate?

Solution

a) For 2002 10x

2.00579 .2579(10) (1 9700) . 3S

4.1 million

For 2003 11x

2.00579 .2579(11) (1 9701) . 3S

4.5 million

b) 23. .00579 .257 95 9 . 703x x

20 .00579 .2579 .9703 3.5x x

20 .00579 .2579 2.5297x x

2.2579 (.2579) 4(.00579)( 2.5297)

2(.00579)x

.2579 .1251.01158

.2579 .3537.01158

x .2579 .3537.01158

x

52.8x 8.3x

According to the model, the number reached 3.5 million in the year 2000. The model closely

matches the graph, so it is accurate

38

Exercise

Cynthia wants to buy a rug for a room that is 20 ft. wide and 27 ft. long. She wants to leave a uniform

strip of floor around the rug. She can afford to buy 170 square feet of carpeting. What dimension should

the rug have?

Solution

x

x

x

x

27 ft

20 ft20 –

2 x

27 – 2 x

The area of the rug is:

27 2 20 2 170x x

2540 54 40 4 170x x x 2540 94 4 170 0x x

24 94 370 0x x Solve for x.

18.5x 5or x

20 2 20 2 5 10 17 2 2 2 5 72 7x and x

Therefore, the dimensions are: 10, 20 ft.

Exercise

Erik finds a piece of property in the shape of a right triangle. He finds that the longer leg is 20 m longer

than twice the length of the shorter leg. The hypotenuse is 10 m longer than the length of the longer leg.

Find the lengths of the sides of the triangular lot.

Solution

: l longer leg

: s shorter leg

Longer leg is 20 m longer than twice the length of the shorter leg

2 20l s

The hypotenuse is 10 m longer than the length of the longer leg

10h l

2 20 10s

2 30s

39

2 2 2l s h

2 2 2(2 20) (2 30)s s s

2 2 24 80 400 4 120 900s s s s s

2 2 24 80 400 4 120 900 0s s s s s

2 40 500 0s s

( 10)( 50) 0s s

10 0s 50 0s

10s 50s

The shorter length is 50 m.

The longer length is 2 20 2(50) 20 120l s

10 120 10 130 h l m

Exercise

An open box is made from a 10-cm by 20-cm piece of tin by cutting a square from each corner and

folding up the edges. The area of the resulting base is 96 2cm . What is the length of the sides of the

squares?

Solution

Area of the base = (10 – 2x)(20 – 2x)

244020200 xxx

200604 2 xx

96200604 2 xx

0104604 2 xx

026152 xx

0)2)(13( xx

202

13013

xx

xx )( 2 onlyx