strategic intervention materials on mathematics 2.0

+ Strategic Intervention

MaterialMathematics

IX

Brian M. Mary Lipay High School

Sta. Cruz, Zambales

Solving Quadratic Equation by Factoring

Approved:

Deomedes M. Eclarino Principal II

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Guide CardLEAST MASTERED SKILLS

Solving Quadratic Equation

Sub Tasks Identifying quadratic equations Rewriting quadratic equations to its

standard form Factor trinomials in the form x2 + bx + c

Determine roots of quadratic equation

ax2 + bx + c = 0, by factoring

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OverviewA quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the following form

ax2 + bx + c = 0,

where a, b, and c are real numbers and a ≠ 0.

Why do you think ‘a’ must

not be equal to

zero in the equation ax2 + bx + c =

0?

How are quadratic equations used in solving real – life problems and in making decisions?

Many formulas used in the physical world are quadratic in nature since they become second-degree equations when solving for one of the variables. Likewise, many word problems require the use of the quadratic equation.

At the enrichment card, we will consider one of the common use of the quadratic equations.

+ Activity Card # 1

__________ 1. 3m + 8 = 15__________ 2. x2 – 5x – 10 = 0 __________ 3. 2t2 – 7t = 12__________ 4. 12 – 4x = 0 __________ 5. 25 – r2 = 4r

Quadratic or Not Quadratic?Direction. Identify which of the following equations are quadratic and which are not. Write QE if the equations are quadratic and NQE if not quadratic equation.

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Activity Card # 2Set Me to Your Standard!

Direction. Write each quadratic equation in standard form, ax2 + bx + c = 0.

1. 3x – 2x2 = 7____________________ 2. 5 – 2r2 = 6r ____________________ 3. 2x(x – 3) = 15 ____________________ 4. (x + 3)(x + 4)= 0 ____________________ 5. (x + 4)2 + 8 = 0 ____________________

+ Activity Card # 3What Made Me?

We learned how to multiply two binomials as follows:

factors(x+2)(x+6) = x2 + 6x + 2x + 12 = x2 + 8x + 12.

termsM u l t i p l y i n g

factorsterms F a c t o r i n gx2 + 8x + 12 = (x + 2)(x + 6)

In factoring, we reverse the operation

The following will enable us to see how a trinomial factors.

x2 + 8x + 12 = (x + 2)(x + 6)

12 = 2 (6)

8 = 2 + 6

Product

Sum

Study Tip

Alternate MethodYou can use the

opposite of FOIL to factor trinomials. For instance, considerExample 1.1

x2+ x – 12

(x + )(x + )

Try factor pair of -12 until the sum of the products of the Inner and Outer terms is x.

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In general, the trinomial x2 + bx + c will factor only if there are two integers, which will we call m and n, such that m + n = b and m(n) = c. Sum

Productm + n m(n)

x2 + bx + c = (x + m)(x + n)

1. a2 + 11a + 18 m + n = 11m(n) = 18 2 + 9 = 11 2(9) = 18

The m and n values are 2 and 9. the factorization is,a2 + 11a + 18 = (x + 2) (x + 9)

2. b2 – 2b – 15 m + n = - 2 m(n) = - 15

3 + (-5) = - 2 3(-5) = - 15The m and n values are 3 and - 5. the factorization

is,b2 – 2b – 15 = (x + 3) (x – 5)

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Factor the following trinomial in the form x2 + bx + c.

x2 + bx + c m + n m(n) (x + m)(x + n)

x2 + 4x – 12 6 + (-2) 6(-2) (x + 6)(x – 2)

w2 – 8w + 12

x2 + 5x - 24

c2 + 6c + 5

r2 + 5r – 14

x2 + 9x + 20

After learning how to factor trinomial in the form x2 + bx + c,we will now determine roots of a quadratic equation using factoring.

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Activity Card # 4 Factor then Solve!

Some quadratic equations can be solved easily by factoring. To solve each equations, the following procedures can be followed.

1. Transform the quadratic equation into standard form if necessary.2. Factor the quadratic expression.3. Set each factor of the quadratic expression equal to 0.4. Solve each resulting equation.

Example. Find the solution of x2 + 9x = -8 by factoring.a. Transform the equation into standard form

x2 + 9x = -8 x2 + 9x + 8 = 0 b. Factor the quadratic expression x2 + 9x + 8 = 0 (x + 1)(x +8) = 0

c. Set each factor equal to 0. (x + 1)(x + 8) = 0 x + 1 =

0 ; x + 8 = 0d. Solve each resulting equation.x + 1 = 0 x + 1 – 1

= 0 -1

x = - 1

x + 8 = 0 x + 8 – 8 = 0 - 8

x = - 8

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Try these!Direction. Determine the roots of the following quadratic equations using factoring.

1. x2 + 8x + 16 = 0 _______________________________________________________________________________________________________________

2. x2 – 9x – 14 = 0 _______________________________________________________________________________________________________________

3. y2 + 9y + 20 = 0 _______________________________________________________________________________________________________________

4. b2 – 10b + 21 = 0 ________________________________________________________________________________________________________________________________________________

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Choose the letter that best answer the question.__________ 1. A polynomial equation of degree 2 that can be written in the form where a, b and c are real numbers, and a

a. Linear Equationb. Linear Inequalityc. Quadratic Equationd. Quadratic Inequality

__________ 2. Which of the following is a quadratic equation?a. c. b. d.

__________ 3. The following are quadratic equation written in standard form except

a. c. b. d.

__________ 4. What is the standard form of the quadratic equation a. c. b. d.

Assessment Card

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__________ 5. What are the factors of the ta. c. b. d.

__________ 6. If one of the factor of the trinomial is what is the other factor?

a. c. b. d.

__________ 7. What is the roots of the quadratic equation ?a. c. b. d.

__________ 8. The roots of the quadratic equation are – 5 and 3. Which of the following quadratic equations has these roots?

a. c. b. d.

__________ 9. Which of the following term must not be equal to 0 in a quadratic equation?

a. b. c. d. __________ 10. In the quadratic equation , one of the roots is 1. What is the other root?

a. b. c. d.

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Enrichment Card*The length of a rectangle is 5 cm more than its width and the area is 50

square cm. Find the length and the width of the rectangle.

Solution: w

w + 5 w + 5

Use the formula and the fact that the area is 50 square cm to set up an algebraic equation.

Simplifying it, we notice that the equation is a quadratic equation.

By using the concepts of solving quadratic equation by factoring, we get

(w + 10) (w – 5) = 0w + 10 = 0 w – 5 = 0w = - 10 w = 5

At this point, we have two possibilities for the width of the rectangle, However, since w = - 10 is impossible to be a width, choose the positive solution, w = 5. Back substitute to find the length,

length, w + 5 = 5 + 5 = 10.Answer: The width is 5 feet and the length is 10 feet.

(Note: It is important to include the correct unit in the presentation of the answer. Make sure to indicate that the width is 5 feet and the length is 10 feet.)

5 cm more than the width

Quadratic Equations in

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Now it’s your turn…!

Problem:The floor of a rectangular room has a length that is 4 feet more

than twice its width. If the total area of the floor is 240 square feet, then find the dimensions of the floor. (Note: The dimensions of the floor is the length and width of the floor.)

Answer:

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Learner’s Material – Mathematics IX, First Edition pp. 27 – 34

Holiday, Berchie. et. al. ALGEBRA 2. USA. The McGraw – Hill Companies, 2008. pp. 253 – 256

Wesner, et. al. ELEMENTARY ALGEBRA with APPLICATIONS. Bernard J. Klein Publishing, 2006 pp. 152 – 156

Reference Card

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Activity Card # 1 Quadratic or Not Quadratic? 1. NQE2. QE3. QE4. NQE5. QE

Answer Card

Activity Card # 2 Set Me to Your Standard1. - 2x2 + 3x – 7 = 0 or 2x2 – 3x + 7 = 02. - 2r2 – 6r + 5 = 0 or 2r2 + 6r – 5 = 03. 2x2 – 6x – 15 = 04. x2 + 7x + 12 = 05. x2 + 8x + 24 = 0

Activity Card # 3 What Made Me? x2 + bx +

cm +

n m(n) (x + m) (x + n)

w2 – 8w + 12 - 6 + 2 -6(2) (w – 6)(w + 2)

x2 + 5x – 24 8 + (-3) 8(-3) (x + 8)(x – 3)

c2 + 6c + 5 5 + 1 5(1) (c + 5)(c + 1)

r2 + 5r – 14 7 + (-2) 7(-2) (r + 7)(r – 2)

x2 + 9x + 20 5 + 4 5(4) (x + 5)(x + 4)

Activity Card # 41. x2 + 8x + 16 = 0 (x + 4)(x + 4) = 0

x + 4 = 0x + 4 – 4 = 0 – 4 x = - 4

2. x2 – 5x – 14 = 0 (x – 7)(x – 2) = 0x – 7 = 0 x – 2 = 0x – 7 + 7 = 0 + 7 x – 2 + 2 = 0 + 2x = 7 x = 2

3. y2 + 9y + 20 = 0 (y + 5)(y + 4) = 0y + 5 = 0 y + 4 = 0 y + 5 – 5 = 0 – 5 y + 4 – 4 = 0 – 4 y = - 5 y = - 4

4. b2 – 10b + 21 = 0 (b – 7)(b – 3) = 0b – 7 = 0 b – 3 = 0b – 7 + 7 = 0 + 7 b – 3 + 3 = 0 + 3b = 7 b = 3

Assessment Card1. c6. d2. a7. a3. a8. d4. d9. a5. c10. a

Enrichment Card Answer: The width is 10 feet and the length is 24 feet.

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