1. Write each expression as a trinomial.
Warm Up
Write each expression as a trinomial.
1. (x – 5)2
x2 – 10x + 25
Factor each expression.
2. x2 –
(x – 9)2

2. Objectives
Solve quadratic equations by completing the square.

3. If a quadratic expression of the form x2 + bx cannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square.

4. Example 1: Completing the Square
Complete the square for the expression. Write the resulting expression as a binomial squared.
x2 – 14x +
Find
x2 – 14x + 49
Add.
(x – 7)2
Factor.

5. Example 2: Completing the Square
Complete the square for the expression. Write the resulting expression as a binomial squared.
x2 + 9x +
Find
Add.
Factor.

6. Example 3
Complete the square for the expression. Write the resulting expression as a binomial squared.
x2 + 4x +
Find
x2 + 4x + 4
Add.
(x + 2)2
Factor.

7. Example 4
Complete the square for the expression. Write the resulting expression as a binomial squared.
x2 – 4x +
Find
x2 – 4x + 4
Add.
(x – 2)2
Factor.

8. Example 5
Complete the square for the expression.
n2 + 4n - 24 = 0
n2 + 4n = 24
n2 + 4n + 4 =
(n + 2)2 = 28

9. CLASSWORK
Worksheet

10. Example 1: Solving a Quadratic Equation by Completing the Square
Solve the equation by completing the square.
x2 = 12x – 20
Collect variable terms on one side.
x2 – 12x = –20
Set up to complete the square.
x2 – 12x = –20 +
Add to both sides.
x2 – 12x + 36 = –
Simplify.

11. Example 1 Continued
(x – 6)2 = 16
Factor.
Take the square root of both sides.
Simplify.
x – 6 = ±4
x – 6 = 4 or x – 6 = –4
Solve for x.
x = 10 or x = 2

12. Example 2: Solving a Quadratic Equation by Completing the Square
Solve the equation by completing the square.
18x + 3x2 = 45
x2 + 6x = 15
Divide both sides by 3.
x2 + 6x = 15 +
Set up to complete the square.
Add to both sides.
x2 + 6x + 9 =
Simplify.

13. Example 2 Continued
(x + 3)2 = 24
Factor.
Take the square root of both sides.
Simplify.

14. Example 3
Solve the equation by completing the square.
x2 – 2 = 9x
Collect variable terms on one side.
x2 – 9x = 2
Set up to complete the square.
x2 – 9x = 2 +
Add to both sides.
Simplify.

15. Take the square root of both sides.
Example 3 Continued
Factor. 9 2
x – 89 4 =
Take the square root of both sides. 9 2 x = 89
Simplify.

16. Example 4
Solve the equation by completing the square.
3x2 – 24x = 27
Divide both sides by 3.
x2 – 8x = 9
Set up to complete the square.
x2 –8x = 9 +
Add to both sides.
Simplify.

17. Example 4 Continued
Solve the equation by completing the square.
Factor.
Take the square root of both sides.
Simplify.
x – 4 =–5 or x – 4 = 5
Solve for x.
x =–1 or x = 9

18. CLASSWORK
Worksheet