1. Chapter 9 Section 3

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by the Quadratic Formula Identify the values of a, b, and c in a quadratic equation. Use the quadratic formula to solve quadratic equations. Solve quadratic equations with only one solution. Solve quadratic equations with fractions. 9.3 2 3 4

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by the Quadratic Formula We can solve any quadratic equation by completing the square, but the method is tedious. In this section, we complete the square on the general quadratic equation ax 2 + bx + c = 0 (where a does not equal 0). By doing this, we get the quadratic formula, which gives the solutions of any quadratic equation. Slide 9.3-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Identify the values of a, b, and c in a quadratic equation. Slide 9.3-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Identify the values of a, b, and c in a quadratic equation. To solve a quadratic equation with the quadratic formula, we must first identify the values of a, b, and c in the standard form of the quadratic equation. The quadratic formula should be memorized and the values of a, b, and c are only determined after the equation is written in standard form. Slide 9.3-5

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write the equation in standard form, if necessary, and then identify the values of a, b, and c. Solution: Slide 9.3-6 EXAMPLE 1 Determining Values of a, b, and c in Quadratic Equations

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Use the quadratic formula to solve quadratic equations. Slide 9.3-7

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. To develop the quadratic formula, we follow the steps given in Section 9.2 for completing the square on ax 2 + bx + c = 0. This formula is also valid, however, for a < 0. Notice that there are two values: one for the + sign and one for the – sign. Use the quadratic formula to solve quadratic equations. Notice that the fraction bar is under –b as well as the radical. When using this formula, be sure to find the values of first. Then divide those results by the value of 2a. Slide 9.3-8

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve 2x 2 + x – 3 = 0. Solution: Slide 9.3-9 EXAMPLE 2 Solving a Quadratic Equation by the Quadratic Formula

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve –x 2 = 8x + 1. Slide 9.3-10 EXAMPLE 3 Rewriting a Quadratic Equation before Solving

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Solve quadratic equations with only one solution. Slide 9.3-11

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve quadratic equations with only one solution. In the quadratic formula, the quantity under the radical, b 2 – 4ac, is called the discriminant. When the discriminant equals 0, the equation has just one rational number solution, and the trinomial ax 2 + bx + c is a perfect square. Slide 9.3-12

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve 9x 2 = 42x – 49. Slide 9.3-13 EXAMPLE 4 Solving a Quadratic Equation with Only One Solution

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Solve quadratic equations with fractions. Slide 9.3-14

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution: Slide 9.3-15 EXAMPLE 5 Solving a Quadratic Equation with Fractions