Solving Quadratic Equations by Using the Quadratic Formula (9-5) Objective: Solve quadratic equations by using the Quadratic Formula. Use the discriminant to determine the number of solutions of a quadratic equation.

Quadratic Formula The solutions of the quadratic equation ax 2 + bx + c = 0, where a ≠ 0, are given by the Quadratic Formula. You can read this formula as “x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a.”

Example 1 Solve x 2 – 2x = 35 by using the Quadratic Formula.  x 2 – 2x – 35 = 0  a = 1  b = -2  c = -35 {7, -5}

Check Your Progress Choose the best answer for the following.  Solve x 2 + x – 30 = 0. Round to the nearest tenth if necessary. A.{6, -5} B.{-6, 5} C.{6, 5} D.  a = 1, b = 1, c = -30

The Quadratic Formula The solutions of quadratic equations are not always integers.

Example 2 Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. a.2x 2 – 2x – 5 = 0  a = 2  b = -2  c = -5 x = {2.2, -1.2}

Example 2 Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary. b.5x 2 – 8x = 4  5x 2 – 8x – 4 = 0  a = 5  b = -8  c = -4 x = {2, -0.4}

Check Your Progress Choose the best answer for the following. A.Solve 5x 2 + 3x – 8 = 0. Round to the nearest tenth if necessary. A.1, -1.6 B.-0.5, 1.2 C.0.6, 1.8 D.-1, 1.4 a = 5, b = 3, c = -8

Check Your Progress Choose the best answer for the following. B.Solve 3x 2 – 6x + 2 = 0. Round to the nearest tenth if necessary. A.-0.1, 0.9 B.-0.5, 1.2 C.0.6, 1.8 D.0.4, 1.6 a = 3, b = -6, c = 2

Example 3 Solve 3x 2 – 5x = 12.  3x 2 – 5x – 12 = 0  a = 3  b = -5  c = -12 x = {3, -1.3}

Check Your Progress Choose the best answer for the following. – Solve 6x 2 + x = 2 by using the Quadratic Formula. A.-0.8, 1.4 B. -2 / 3, 1 / 2 C. -4 / 3, 1 D.0.6, 2.2 a = 6, b = 1, c = -2 6x 2 + x – 2 = 0

The Discriminant In the quadratic formula, the expression inside the radical (b 2 – 4ac) is called the discriminant. The discriminant can be used to find the number of solutions of the quadratic equation. – If b 2 – 4ac is positive, then the equation has two solutions. – If b 2 – 4ac is zero, then the equation has one solution. – If b 2 – 4ac is negative, then the equation has no real solution.

Example 4 State the value of the discriminant of 3x x = 12. Then determine the number of real solutions of the equation.  3x x – 12 = 0  a = 3  b = 10  c = -12 (10) 2 – 4(3)(-12)= 244 Two Solutions

Check Your Progress Choose the best answer for the following. – State the value of the discriminant for the equation x 2 + 2x + 2 = 0. Then determine the number of real solutions of the equation. A.-4; no real solutions B.4; 2 real solutions C.0; 1 real solution D.Cannot be determined (2) 2 – 4(1)(2) a = 1, b = 2, c = 2