5.9 The Quadratic Formula 12/11/2013

Quadratic Formula: Is used to solve quadratic equations that cannot be solved using the “Big X” method. It can also be used instead of the “Big X”. The equation must be written as: ax2 + bx +c = 0.

Example 1 Solve the equation . - x 2 + 2x 15 = 0 = ( ) 3 x – 5 + = 3 x Factor. = ( ) 3 x – 5 + (mult) -15 = 3 x – or 5 + 5 -3 + 3 = + 3 - 5 = - 5 2 (add) = 3 x 5 – = x 2a b – + 4ac b 2 a = 1 b = 2 c = -15 𝑥= −2+8 2 = 6 2 =3 = x 2 – + 4 22 ( ) 1 15 𝑥= −2−8 2 = −10 2 =−5 𝑥= −2± 4+60 2 = −2± 64 2 = −2±8 2

Example 2 = x 2a b – + 4ac b 2 Solve . = x 2 + 3x 5 – SOLUTION = x 3 – Solve an Equation with Two Real Solutions = x 2a b – + 4ac b 2 Solve . = x 2 + 3x 5 – SOLUTION = x 3 – + 4 32 ( ) 1 5 2 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 4.19

Use the quadratic formula to solve the equation. Checkpoint Solve an Equation with Two Real Solutions Use the quadratic formula to solve the equation. 1. = x 2 + 2x 3 – 3. = 3x 2 2x 6 – ANSWER 3, – 1 ANSWER 1 19 3 ≈ 1.79 + 1.12 – 2. = 2x 2 + 4x 1 – ANSWER 2 – + 6 ≈ 0.22, 2.22

Example 3 2a x = – b + b 2 4ac Solve x 2 2x 2 + 0. = SOLUTION x = + – Solve an Equation with Imaginary Solutions 2a x = – b + b 2 4ac Solve x 2 2x 2 + 0. = SOLUTION x = + – 2 1 ( 4 22 Substitute values in the quadratic formula: a 1, b 2, and c 2. x = + – 2 4 Simplify. – 2 + – 2i Simplify and rewrite using the imaginary unit i. x = 2 Simplify. x = + – 1 i + – 1 i ANSWER The solutions are and . 6

Use the quadratic formula to solve the equation. Checkpoint Use the Quadratic Formula Use the quadratic formula to solve the equation. ANSWERS 4. x 2 4x 5 + = + – 2 i x 2 2x 3 + = – 5. + – 1 i 2 6. 2x 2 x = 4 – + + – 1 i 4 31

Homework: 5.9 p.278 #31-40