1. 5.9 The Quadratic Formula 1/30/2013

2. Quadratic Formula: Is used to solve quadratic equations that cannot be solved using the “Big X” method. The equation must be written as: ax 2 + bx +c = 0.

3. Example 1 Solve an Equation with Two Real Solutions Solve. = x 2x 2 +3x3x5 – 0 SOLUTION = x 2a2a b – + – 4ac4acb 2b 2 – = x 3 – + – 43232 – () 1 () 5 – 2 () 1 Substitute values in the quadratic formula: a 1, b 3, and c 5. – = == = x 3 – + – 29 2 Simplify. ANSWER The solutions are and. 3 – + 29 2 ≈ 1.19 3 –– 29 2 ≈ 4.19 –

4. Checkpoint Use the quadratic formula to solve the equation. Solve an Equation with Two Real Solutions 1. = x 2x 2 +2x2x3 – 0 2. = 2x 22x 2 +4x4x1 – 0 ANSWER 2 – + 6 2 ≈ 0.22, 2 – 6 2 ≈ 2.22 – – ANSWER 3,3, – 1 3. = 3x 23x 2 2x2x6 – 0 – ANSWER 119 3 ≈ 1.79 + 119 3 ≈ 1.12 – –

5. Example 2 Solve an Equation with One Real Solution Solve x 2x 2 6x6x – = – 9.9. SOLUTION x 2x 2 6x6x9 – + = 0 Write equation in standard form. 2a2a x = – b + – b 2b 2 – 4ac4ac 6 2 Simplify. x = + – 6 0 2 x = 3 ANSWER The solution is 3. x = + – ( – 6 ( – ( – ( – 1 9)9) ( ( ( 4 1 ( ( 2 Substitute values in the quadratic formula: a = 1, b = - 6, and c = 9.

6. Example 3 Solve an Equation with Imaginary Solutions Solve x 2x 2 2x2x2 ++ 0.0. = x = + – – 2 – 1 2 ( (( ( 4 1 ( ( 2 2 Substitute values in the quadratic formula: a 1, b 2, and c 2. = == 2a2a x = – b + – b 2b 2 – 4ac4ac x = + – – 2 – 4 2 Simplify. Simplify and rewrite using the imaginary unit i. x = + – – 2 2 2i2i + – 1 i ANSWER The solutions are and. – 1 i – Simplify. x = + – – 1 i SOLUTION

7. Checkpoint Use the quadratic formula to solve the equation. Use the Quadratic Formula 4. x 2x 2 4x4x5 ++ 0 = + – – 2 i x 2x 2 2x2x3 + 0 = –– 5. + – 1 i 2 6. 2x 22x 2 x = 4 – + + – 1 i 4 31 ANSWERS

8. Homework: 5.9 p.278 #31-40