1. Completing the Square

2. Learning Intention/Success Criteria
LI: We are learning to factor quadratics by completing the square SC: I know how to -take the square root of values -solve for the roots -explain why completed square form is also known as vertex form -solve quadratic equations by completing the square -divide and add integers -simplify equations and expressions -factor quadratics using stick man

3. Example 1: List the vertex of the equation
h(x) = -4(x – 5)2 + 9
(5, 9)

4. Guided Practice 1
Find the vertex of the following equation: f(x) = 2(x + 6)2 – 8
A] (6, 8)
B] (6, - 8)
C] (-6, -8)
D] (-6, 8)

5. Guided Practice 2
Find the vertex of the following equation: d(x) = 7(x – 12)2 – 3
A] (12, 3)
B] (12, - 3)
C] (-12, -3)
D] (-12, 3)

6. Guided Practice 3
Find the vertex of the following equation: j(x) = -3(x + 8)2 + 5
A] (8, 5)
B] (8, - 5)
C] (-8, -5)
D] (-8, 5)

7. Steps for Completing the Square
Divide all terms by A
Set the equation equal to C
Divide B by 2 and square it
Add the value of Step 3 to both sides of the equation
Simplify
Factor the left side (called completed square form)
Square root both sides
Solve each equation, getting two answers. Round your answer to the nearest tenth

8. Example 2: Solve for x by completing the square x2 + 4x – 1 = 0
______________
+ 1
+ 1
x2 + 4x = 1
𝐵 2 2 = 3
= 2 2
= 4
x2 + 4x + 4 = 1 + 4 4
x2 + 4x + 4 = 5 5
(x + 2)2 = 5 6
(x + 2)2 = 5 7
𝑥+2= ± 5 8
𝑥=−2± 5

9. Guided Practice 4
Solve for x by completing the square: x2 + 8x – 3 = 0

10. Guided Practice 4
Solve for x by completing the square: x2 + 8x – 3 = 0
A] x =  1
B] x = -4  3
C] x = 0
D] x = -4  19

11. Guided Practice 4
Solve for x by completing the square: x2 + 8x – 3 = 0
𝐵 2 2 =
= 4 2
= 16
+ 3
+ 3
________________
x2 + 8x = 3
________________
+ 16
+ 16
x2 + 8x + 16 = 19
(x + 4)2 =
x + 4 = ± 19
________________
- 4
- 4
x = - 4 ± 19

12. Guided Practice 5
Solve for x by completing the square: x2 + 24x + 32 = 0

13. Guided Practice 5
Solve for x by completing the square: x2 + 24x + 32 = 0
A] −12 ±4 7
B] 12 ±4 7
C] −12 ±4 11
D] 12 ±4 11

14. Guided Practice 4
Solve for x by completing the square: x2 + 24x + 32 = 0 =
𝐵 2 2
= 12 2
= 144
________________
-32
-32
x2 + 24x = -32
x = -12 ± 16∗7
________________
+ 144
+ 144
x = -12 ±4 7
x2 + 24x +144 = 112
(x + 12)2 = 112
x + 12 = ± 112
________________
- 12
- 12
x = -12 ± 112

15. Example 2: Solve for x by completing the square
4x2 + 16x – 9 = 0
4x2 + 16x – 9 = 0
___ 4
___ 4
___ 4
x2 + 4x – 𝟗 𝟒 = 0
+ 𝟗 𝟒
+ 𝟗 𝟒
_________________
x2 + 4x = 𝟗 𝟒
= 𝟒 𝟐 𝟐
𝑩 𝟐 𝟐
= 𝟐 𝟐
= 4
x2 + 4x = 𝟗 𝟒
_________________
+ 4
+ 4
x2 + 4x + 4 = 𝟗 𝟒 + 𝟏𝟔 𝟒

16. x2 + 4x + 4 = 𝟐𝟓 𝟒
(x + 2)2 = 𝟐𝟓 𝟒
x =  𝟓 𝟐
_________________
- 2
- 2
x = -2  𝟓 𝟐
x = 𝟓 𝟐 x = 𝟓 𝟐
x = 1/2
x = -4.5

17. Guided Practice 6
Solve for x by completing the square: 7x2 + 28x – 49 = 0

18. Poll Solve for x by completing the square: x2 + 24x + 32 = 0 A] −2 ± 3
D] −2 ± 11

19. Guided Practice 6
Solve for x by completing the square: 7x2 + 28x – 49 = 0
___ 7
___ 7
___ 7 =
𝐵 2 2
= 2 2
= 4
x2 + 4x – 7 = 0
______________
+ 7
+ 7
(x + 2)2 = 11
x2 + 4x = 7
x + 2 = ± 11
+ 4
+ 4
______________
- 2
- 2
x2 + 4x + 4 = 11
x = −2 ± 11
(x + 2)2 = 11