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6.9

Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula

Slide 6 - 2Copyright © 2009 Pearson Education, Inc.

FOIL

A binomial is an expression that contains two terms.

To multiply two binomials, we use the FOIL method.

F = FirstO = OuterI = InnerL = Last

(a + b)(c + d) = ac + ad + bc + bd

F

I

O

L

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Example

Multiply: (2x + 4)(x + 6)

644622 xxxx

244122 2 xxx

24162 2 xx

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Practice Problem: (x + 7)(x + 2)

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To Factor Trinomial Expressions of the Form x2 + bx + c Find two numbers whose product is c and

whose sum is b. Write the factors in the form

(x + ) (x + )

Check your answer by multiplying the factors using the FOIL method.

One number from step 1

Other number from step 1

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Factoring Example

Factor x2 7x + 12. We need to find two numbers whose product is

12 and whose sum is 7.

(x 3)(x 4)

3 + 4 = 7

2 + 6 = 8

1 + 12 = 13

Sum of Factors

(3)(4)

(2)(6)

1(12)

Factors of 12

3 + 4 = 7(3)(4)

2 + 6 = 8(2)(6)

1 + 12 = 131(12)

Sum of FactorsFactors of 12

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Practice Problem: Factor

122 xx

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Factoring Trinomials of the Form ax2 + bc + c, a 1.

Write all pairs of factors of the coefficient of the squared term, a.

Write all pairs of the factors of the constant, c. Try various combinations of these factors until

the sum of the products of the outer and inner terms is bx.

Check your answer by multiplying the factors using the FOIL method.

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Example: Factoring

Factor 3x2 + 14x + 8. (3x + )(x + )

Thus, 3x2 + 14x + 8 = (3x + 2)(x + 4).

14x Correct middle term(3x + 2)(x + 4)

10x(3x + 4)(x + 2)

11x(3x + 8)(x + 1)

25x(3x + 1)(x + 8)

Sum of Outer and Inner Terms

Possible Factors

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Practice Problem: Factor

1572 2 xx

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Solving Quadratic Equations by Factoring

Standard Form of a Quadratic Equation

ax2 + bx + c = 0, a 0

Zero-Factor Property

If a • b = 0, then a = 0 or b = 0.

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To Solve a Quadratic Equation by Factoring

Use the addition or subtraction property to make one side of the equation equal to 0.

Factor the side of the equation not equal to 0. Use the zero-factor property to solve the

equation.

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Example: Solve by Factoring

Solve 4x2 + 17x 15 = 0.

The solutions are 5 and ¾.

0534 xx

034 x 05x34 x 5x

4

3x

or

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Practice Problem: Solve by factoring

07236 2 xx

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Quadratic Formula

For a quadratic equation in standard form, ax2 + bx + c = 0, a 0, the quadratic formula is

2 4

2

b b acx

a

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Example: Using the Quadratic Formula

Solve the equation 3x2 + 2x 7 = 0.

a = 3, b = 2 and

c = 7

2 4

2

b b acx

a

32

73422 2

6

8442

6

882

6

2222

3

221

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Practice Problem: solve using the quadratic equation

0153 2 xx

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Homework: P 360 # 7 – 34