5.7 Completing the Square 2011
January 07, 2011
5‐7 Completing the Square
Objectives:
*Solve equations by completing the square.
*Rewrite functions by completing the square.
Dec 1­1:26 PM
1
5.7 Completing the Square 2011
January 07, 2011
Check Skills You'll Need:
Simplify each expression
1. (x ‐ 3)(x ‐ 3)
4. ±√25
2. (2x ‐ 1)(2x ‐ 1)
5. ±√48
6. ±√‐4
3. (x + 4)(x + 4)
7. ±√9/16
Dec 1­2:09 PM
2
5.7 Completing the Square 2011
January 07, 2011
You can solve an equation in which one side is a perfect square
trinomial by finding square roots.
Example #1: Solve x2 + 10x + 25 = 36
x2 + 10x + 25 = 36
(x + 5)2 = 36
x + 5 = ±6
x+5=6
or
x + 5 = ‐6
x=1
x = ‐11
Dec 1­2:16 PM
3
5.7 Completing the Square 2011
January 07, 2011
If one side of an equation is not a perfect square trinomial, you can rewrite
the constant term to get a perfect square trinomial. The process of finding
the last term of a perfect square trinomial is called completing the square.
Use the following relationship to find the term that will complete the
square.
x2 + bx + ( b2 )2 = (x + b2 )2
Dec 2­9:44 PM
4
5.7 Completing the Square 2011
January 07, 2011
Example #2: Find the missing value to complete the square.
x2 ‐ 8x +
(
b
2
)2 = ( ‐82 )2 = 16
x2 ‐ 8x + 16
Dec 2­9:44 PM
5
5.7 Completing the Square 2011
January 07, 2011
Example #3: Find the missing value to complete the square.
x2 + 7x +
(
b
2
)2 = (
7 2
)2 = 12.25
x2 + 7x + 12.25
Dec 2­9:44 PM
6
5.7 Completing the Square 2011
January 07, 2011
Jan 5­10:32 AM
7
5.7 Completing the Square 2011
January 07, 2011
Jan 5­10:34 AM
8
5.7 Completing the Square 2011
January 07, 2011
You can solve any quadratic equation by completing the square.
Example #4: Solve x2 ‐ 12x + 5 = 0
( ‐122 )2 = 36
x2 ‐ 12x = ‐5
x2 ‐ 12x + 36 = ‐5 + 36
(x ‐ 6)2 = 31
x ‐ 6 = ±√31
x = 6 ±√31
Find
(
b
2
)2
Rewrite so all terms containing x are on one side.
Complete the square by adding 36 to each side.
Factor the perfect square trinomial.
Find square roots
Solve for x.
Dec 2­9:45 PM
9
5.7 Completing the Square 2011
January 07, 2011
Example #5: Solve each equation by completing the square.
a. x2 + 4x ‐ 4 = 0
b. x2 ‐ 2x ‐ 1 = 0
Dec 2­9:45 PM
10
5.7 Completing the Square 2011
January 07, 2011
By completing the square, you can solve equations that cannot by
solved by factoring, finding square roots, or graphing.
Example #6: Solve x2 ‐ 8x + 36 = 0
(
‐8
2
)2 = 16
x2 ‐ 8x = ‐36
x2 ‐ 8x + 16 = ‐36 + 16
(x ‐ 4)2 = ‐20
x ‐ 4 = ±√‐20
x = 4 ± 2i√5
Dec 2­9:45 PM
11
5.7 Completing the Square 2011
January 07, 2011
Example #7: Write y = x2 + 6x + 2 in vertex form.
x2 + 6x + 2 = 0
(
6
2
)2 = 9
x2 + 6x = ‐2
x2 + 6x + 9 = ‐2 + 9
(x + 3)2 = 7
2
(x + 3) ‐ 7 = 0
y = (x + 3)2 ‐ 7
What is the
vertex of the
function?
(‐3, ‐7)
Dec 2­9:47 PM
12
5.7 Completing the Square 2011
January 07, 2011
Example #8: Write each equation in vertex form.
a. y = x2 ‐ 10x ‐ 2
b. y = x2 + 5x + 3
Dec 2­9:48 PM
13
5.7 Completing the Square 2011
January 07, 2011
Homework:
page 285 (1 ‐ 18, 28, 31)
Dec 2­10:44 PM
14