1. Solving Quadratic Equations

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

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

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.

5. 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

6. 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

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

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

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

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

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