Solving Quadratic Equations

Solving a Quadratic Equation by taking the Square Root 11x2 + 3 = 36 11x2 = 33 x2 = 3 x = + √3

Solving a Quadratic Equation by Factoring 3x2 + 10x – 8 = 0 (3x – 2)(x + 4) = 0 3x – 2 = 0 and x + 4 = 0 x = 2/3 and x = -4

How can you solve a quadratic equation when you can’t take the square root or factor? We use a process known as completing the Square.

Solving Quadratic Equations by Completing the Square Completing the Square is a process where you force a quadratic expression to factor Look at a perfect square trinomial x2 + 8x + 16 = (x + 4)2 Notice that ½(8) = 4 and 42 = 16

x2 + 6x + 9 = (x + 3)2 Notice ½(6) = 3 and 32 = 9 This demonstrates a pattern for perfect square trinomials x2 + bx + c = x2 + bx + (b/2)2 = (x + b/2)2

Completing the Square x2 – 6x + c Find ½(b) 1. ½(-6) = -3 Square ½(b) 2. (-3)2 = 9 Write the perfect 3. x2 – 6x + 9 square trinomial 4. Factor the trinomial 4. (x – 3)2

Try Completing the Square x2 + 14x + c ½(14) = 7 72 = 49 x2 + 14x + 49 (x + 7)2

Try: x2 + 3/2x + c ½(3/2) = ¾ (3/4)2 = 9/16 x2 + 3/2x + 9/16 (x + 3/2)2

Using Completing the Square to solve a Quadratic Equation x2 – 16x + 8 = 0 x2 – 16 = -8 x2 – 16 + 64 = -8 + 64 (x – 8)2 = 56 x – 8 = + √56 x = 8 + √56 x = 8 + 2√14

Example 3x2 + 6x + 15 = 0 x2 + 2x + 5 = 0 x2 + 2x = -5 x2 + 2x + 1 = -5 + 1 (x + 1)2 = - 4 x + 1 = + √-4 x + 1 = + 2i x = - 1 + 2i