1. Slide 6 - 1 Copyright © 2009 Pearson Education, Inc. 6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula

2. Slide 6 - 2 Copyright © 2009 Pearson Education, Inc. FOIL A binomial is an expression that contains two terms. To multiply two binomials, we use the FOIL method. F = First O = Outer I = Inner L = Last (a + b)(c + d) = ac + ad + bc + bd F I O L

3. Slide 6 - 3 Copyright © 2009 Pearson Education, Inc. Example Multiply: (2x + 4)(x + 6)

4. Slide 6 - 4 Copyright © 2009 Pearson Education, Inc. Practice Problem: (x + 7)(x + 2)

5. Slide 6 - 5 Copyright © 2009 Pearson Education, Inc. To Factor Trinomial Expressions of the Form x 2 + 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

6. Slide 6 - 6 Copyright © 2009 Pearson Education, Inc. Factoring Example Factor x 2  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 =  13 1(12) Sum of FactorsFactors of 12

7. Slide 6 - 7 Copyright © 2009 Pearson Education, Inc. Practice Problem: Factor

8. Slide 6 - 8 Copyright © 2009 Pearson Education, Inc. Factoring Trinomials of the Form ax 2 + 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.

9. Slide 6 - 9 Copyright © 2009 Pearson Education, Inc. Example: Factoring Factor 3x 2 + 14x + 8. (3x + )(x + ) Thus, 3x 2 + 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

10. Slide 6 - 10 Copyright © 2009 Pearson Education, Inc. Practice Problem: Factor

11. Slide 6 - 11 Copyright © 2009 Pearson Education, Inc. Solving Quadratic Equations by Factoring Standard Form of a Quadratic Equation ax 2 + bx + c = 0, a  0 Zero-Factor Property If a b = 0, then a = 0 or b = 0.

12. Slide 6 - 12 Copyright © 2009 Pearson Education, Inc. 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.

13. Slide 6 - 13 Copyright © 2009 Pearson Education, Inc. Example: Solve by Factoring Solve 4x 2 + 17x  15 = 0. The solutions are  5 and ¾. or

14. Slide 6 - 14 Copyright © 2009 Pearson Education, Inc. Practice Problem: Solve by factoring

15. Slide 6 - 15 Copyright © 2009 Pearson Education, Inc. Quadratic Formula For a quadratic equation in standard form, ax 2 + bx + c = 0, a  0, the quadratic formula is

16. Slide 6 - 16 Copyright © 2009 Pearson Education, Inc. Example: Using the Quadratic Formula Solve the equation 3x 2 + 2x  7 = 0. a = 3, b = 2 and c =  7

17. Slide 6 - 17 Copyright © 2009 Pearson Education, Inc. Practice Problem: solve using the quadratic equation

18. Slide 6 - 18 Copyright © 2009 Pearson Education, Inc. Homework: P 360 # 7 – 34