Chapter 9 Section 3

Solving Quadratic Equations by the Quadratic Formula 9.3 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. 2 3 4

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 ax2 + 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

Identify the values of a, b, and c in a quadratic equation. Objective 1 Identify the values of a, b, and c in a quadratic equation. Slide 9.3-4

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

EXAMPLE 1 Determining Values of a, b, and c in Quadratic Equations Write the equation in standard form, if necessary, and then identify the values of a, b, and c. Solution: Slide 9.3-6

Use the quadratic formula to solve quadratic equations. Objective 2 Use the quadratic formula to solve quadratic equations. Slide 9.3-7

Use the quadratic formula to solve quadratic equations. To develop the quadratic formula, we follow the steps given in Section 9.2 for completing the square on ax2 + 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. 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

EXAMPLE 2 Solve 2x2 + x – 3 = 0. Solution: Solving a Quadratic Equation by the Quadratic Formula Solve 2x2 + x – 3 = 0. Solution: Slide 9.3-9

Rewriting a Quadratic Equation before Solving EXAMPLE 3 Rewriting a Quadratic Equation before Solving Solve –x2 = 8x + 1. Solution: Slide 9.3-10

Solve quadratic equations with only one solution. Objective 3 Solve quadratic equations with only one solution. Slide 9.3-11

Solve quadratic equations with only one solution. In the quadratic formula, the quantity under the radical, b2 – 4ac, is called the discriminant. When the discriminant equals 0, the equation has just one rational number solution, and the trinomial ax2 + bx + c is a perfect square. Slide 9.3-12

Solving a Quadratic Equation with Only One Solution EXAMPLE 4 Solving a Quadratic Equation with Only One Solution Solve 9x2 = 42x – 49. Solution: Slide 9.3-13

Solve quadratic equations with fractions. Objective 4 Solve quadratic equations with fractions. Slide 9.3-14

Solving a Quadratic Equation with Fractions EXAMPLE 5 Solving a Quadratic Equation with Fractions Solve. Solution: Slide 9.3-15