1. Chapter 6 Section 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Solving Quadratic Equations by Factoring Solve quadratic equations by factoring. Solve other equations by factoring. 1 1 2 26.56.5

3. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Quadratic Equations by Factoring. A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0. Slide 6.5 - 3 The form ax 2 + bx + c = 0 is the standard form of a quadratic equation. For example, and are all quadratic equations, but only x 2 + 5x +6 = 0 is in standard form. Until now, we have factored expressions, including many quadratic expressions. In this section we see how we can use factored quadratic expressions to solve quadratic equations.

4. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Solve quadratic equations by factoring. Slide 6.5 - 4

5. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We use the zero-factor property to solve a quadratic equation by factoring. If a and b are real numbers and if ab = 0, then a = 0 or b = 0. That is, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must, but both may be 0. Solve quadratic equations by factoring. Slide 6.5 - 5

6. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solve. Solution: Using the Zero-Factor Property Slide 6.5 - 6 or

7. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. EXAMPLE 2 Solution: Solving Quadratic Equations Slide 6.5 - 7 or

8. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve quadratic equations by factoring. (cont’d) Slide 6.5 - 8 In summary, follow these steps to solve quadratic equations by factoring. Step 1: Write the equation in standard form— that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side. Step 2: Factor completely. Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the resulting equations. Step 4: Check each solution in the original equation.

9. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solution: Solving a Quadratic Equation with a Common Factor Slide 6.5 - 9 Solve 3m 2 − 9m = 30. A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions.

10. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Solving Quadratic Equations Slide 6.5 - 10 Solve.

11. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Solving Quadratic Equations (cont’d) Slide 6.5 - 11 Solve.

12. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Solve other equations by factoring. Slide 6.5 - 12

13. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solve. Solution: Solving Equations with More than Two Variable Factors Slide 6.5 - 13

14. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solve. Solving Equations with More than Two Variable Factors (cont’d) Slide 6.5 - 14 Solution:

15. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. EXAMPLE 6 Solving an Equation Requiring Multiplication before Factoring Slide 6.5 - 15 Solution: