1. Completing the Square

2. What do you get when you foil the following expressions?
(x + 1) (x+1)=
(x + 6)2 =
(x + 7)2 =
(x + 2) (x+2) =
(x + 8)2 =
(x + 3) (x+3) =
(x + 4) (x+4) =
(x + 9)2 =
(x + 5) (x+5) =
(x + 10)2 =

3. What do you get when you foil the following expressions?
x2 + 2x + 1
(x + 10)2 =
x2 + 20x + 100
(x + 2)2 =
x2 + 4x + 4
(x - 13)2 =
x2 - 26x + 169
(x - 3)2 =
x2 - 6x + 9
(x - 25)2 =
x2 - 50x + 625
(x - 4)2 =
x2 - 8x + 16
(x – 0.5)2 =
x2 - x
x2 – 6.4x
(x + 5)2 =
x2 + 10x + 25
(x – 3.2)2 =

4. Fill in the missing number to complete a perfect square.
x2 + 2x + ____
x2 - 14x + ___
x2 + 8x + ___
x2 – 20x + ___
x2 + 6x + ___
x2 + 16x + _____

5. Fill in the missing number to complete a perfect square.
x2 + 10x + 25
x2 + 10x + ___
= (x + 5)2
x2 - 30x + 225
x2 - 30x + ___
= (x - 15)2
x2 – 2.8x
x2 – 2.8x + ___
= (x – 1.4)2
x2 + 18x + 81
x2 + 18x + ___
= (x + 9)2
x2 + 12x + ___
x2 + 12x + 36
= (x + 6)2
x x + _____
x x
= (x – 0.25)2

6. Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 + 14x - 10
y = x2 + 14x + ____ - 10
y = x2 + 14x
y = (x + 7)2 -59
The vertex is at (-7, -59)

7. Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 - 12x + 5
y = x2 - 12x + ____ + 5
y = x2 - 12x
y = (x - 6)2 - 31
The vertex is at (6, -31)

8. Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 - 28x + 200
The vertex is at (14, 4)
y = x2 - 28x + ____ + 200
y = x2 - 28x
y = (x - 14)2 + 4

9. Changing from standard form to vertex form
By completing the square on a quadratic in standard form, it is changed into vertex form
Change to vertex form:
y = x2 – 0.75x - 1
y = x2 – 0.75x + ____ + - 1
y = x2 – 0.75x
The vertex is at (0.375, )
y = (x – 0.375)2 –

10. Change to vertex form:
y = x2 + 4x + 10
y = x2 + 4x + ___
y = x2 + 4x
y = (x + 2)2 + 6

11. Change to vertex form:
y = x2 + 19x - 1
y = x2 + 19x + ___ - 1
y = x2 + 19x – 90.25
y = (x + 9.5)

12. More Complicated Versions of Completing the Square
If the leading coefficient is not equal to 1, completing the square is slightly more difficult.
Directions for Completing the Square:
1.) Move the constant out of the way.
2.) Factor out A from the x2 and x term.
3.) Determine what is half of the remaining B.
4.) Square it and put this in for C.
5.) Put in a constant to cancel out the last step.
6.) Write the parenthesis as a perfect square and simplify everything else.

13. Change to vertex form: y = 2x2 + 4x + 10
Vertex at (-1, 8)

14. Change to vertex form: y = 3x2 + 12x + 22
Vertex at (-2, 10)
y = 3(x + 2)2 + 10

15. Change to vertex form:
y = 6x2 - 48x + 65

16. Change to vertex form:
y = 7x2 - 98x + 400

17. Change to vertex form:
y = 12x2 - 60x + 312

18. Change to vertex form: y = -5x2 + 20x - 32
Vertex at (2, -12)
y = -5(x - 2)2 - 12

19. Change to vertex form: y = -6x2 + 72x - 53
Vertex at (6, 163)
y = -6(x2 - 12x + 36)
y = -6(x - 6)

20. Methods of Locating the Vertex of a Parabola:
If the quadratic is in vertex form:
𝑦=𝑎 𝑥−ℎ 2 +𝑘
The vertex (h, k):
If the quadratic is in factored form:
The x value of the vertex is halfway between the roots. Plug in & solve to find the y value.
𝑦=𝑎 𝑥−__ 𝑥−__
If the quadratic is in standard form:
Complete the square to change to vertex form.
𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐

21. Change to vertex form:
Vertex at (-0.3, -2.45)

22. Change to vertex form:

23. Change to vertex form:

24. Change to vertex form:

25. Solve by completing the square.

26. Solve by completing the square.

27. Example: Solve by completing the square: x2 + 6x – 8 = 0

28. Solve by completing the square:

29. Solve by completing the square:

30. Solve by completing the square:
This is called the Quadratic Formula.
You must memorize it!!!