1. 9-8 Completing the Square Warm Up Lesson Presentation Lesson Quiz
Holt Algebra 1

2. Warm Up
Simplify. 1. 2. 19 3. 4.

3. Warm Up
Solve each quadratic equation by factoring.
5. x2 + 8x + 16 = 0
6. x2 – 22x = 0
7. x2 – 12x + 36 = 0
x = –4
x = 11
x = 6

4. Objective
Solve quadratic equations by completing the square.

5. An expression in the form x2 + bx is not a perfect square
An expression in the form x2 + bx is not a perfect square. However, you can use the algorithm below to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

6. Example 1: Completing the Square
Complete the square to form a perfect square trinomial.
A. x2 + 2x +
B. x2 – 6x +
x2 + 2x
x2 + –6x
Identify b. .
x2 + 2x + 1
x2 – 6x + 9

7. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
a. x2 + 12x +
b. x2 – 5x +
x2 + 12x
x2 + –5x
Identify b. .
x2 – 6x +
x2 + 12x + 36

8. Check It Out! Example 1
Complete the square to form a perfect square trinomial.
c. 8x + x2 +
x2 + 8x
Identify b. .
x2 + 12x + 16

9. To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

10. Solving a Quadratic Equation by Completing the Square

11. Example 2A: Solving x2 +bx = c
Solve by completing the square.
x2 + 16x = –15
The equation is in the form x2 + bx = c.
Step 1 x2 + 16x = –15
Step 2 .
Step 3 x2 + 16x + 64 = –
Complete the square.
Step 4 (x + 8)2 = 49
Factor and simplify.
Take the square root of both sides.
Step 5 x + 8 = ± 7
Step 6 x + 8 = 7 or x + 8 = –7
x = –1 or x = –15
Write and solve two equations.

12. Example 2B: Solving x2 +bx = c
Solve by completing the square.
x2 – 4x – 6 = 0
Write in the form x2 + bx = c.
Step 1 x2 + (–4x) = 6
Step 2 .
Step 3 x2 – 4x + 4 = 6 + 4
Complete the square.
Step 4 (x – 2)2 = 10
Factor and simplify.
Take the square root of both sides.
Step 5 x – 2 = ± √10
Step 6 x – 2 = √10 or x – 2 = –√10
x = 2 + √10 or x = 2 – √10
Write and solve two equations.

13. Check It Out! Example 2a
Solve by completing the square.
x2 + 10x = –9
The equation is in the form x2 + bx = c.
Step 1 x2 + 10x = –9
Step 2 .
Step 3 x2 + 10x + 25 = –9 + 25
Complete the square.
Factor and simplify.
Step 4 (x + 5)2 = 16
Take the square root of both sides.
Step 5 x + 5 = ± 4
Step 6 x + 5 = 4 or x + 5 = –4
x = –1 or x = –9
Write and solve two equations.

14. Example 3A: Solving ax2 + bx = c by Completing the Square
Solve by completing the square.
–3x2 + 12x – 15 = 0
Step 1
Divide by – 3 to make a = 1.
x2 – 4x + 5 = 0
x2 – 4x = –5
Write in the form x2 + bx = c.
x2 + (–4x) = –5
Step 2 .
Step 3
x2 – 4x + 4 = –5 + 4
Complete the square.

15. Example 3A Continued
Solve by completing the square.
Step 3
x2 – 4x + 4 = –5 + 4
–3x2 + 12x – 15 = 0
Step 4
(x – 2)2 = –1
Factor and simplify.
There is no real number whose square is negative, so there are no real solutions.

16. Lesson Quiz: Part I
Complete the square to form a perfect square trinomial.
1. x2 +11x +
2. x2 – 18x +
Solve by completing the square.
3. x2 – 2x – 1 = 0
4. 3x2 + 6x = 144
5. 4x x = 23 81
6, –8