Completing the Square
1
What do you get when you foil the following
expressions?
(x + 1) (x+1)=
(x + 6)2 =
(x + 2) (x+2) =
(x + 7)2 =
(x + 3) (x+3) =
(x + 8)2 =
(x + 4) (x+4) =
(x + 9)2 =
(x + 5) (x+5) =
(x + 10)2 =
2
What do you get when you foil the following
expressions?
(x + 1)2 = 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 + 0.25
(x + 5)2 = x2 + 10x + 25
2
(x – 3.2)2 = x – 6.4x + 10.24
3
Fill in the missing number to complete a
perfect square.
x2 + 2x + ____
x2 - 14x + ___
x2 + 8x + ___
x2 – 20x + ___
x2 + 6x + ___
x2 + 16x + _____
4
Fill in the missing number to complete a
perfect square.
x2 + 10x + ___
25
x2
+ 18x + ___
81
x2 + 12x + ___
36
= (x + 5)2
= (x +
9)2
= (x + 6)2
x2 - 30x + ___
225
= (x - 15)2
x2 – 2.8x + 1.96
___ = (x – 1.4)2
x2 + 0.5x + _____
0.0625 = (x – 0.25)2
5
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 + 49 - 10 - 49
y = (x + 7)2 -59
The vertex is
at (-7, -59)
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 - 12x + 5
y = x2 - 12x + ____ + 5
y = x2 - 12x + 36
y = (x - 6)2 - 31
+ 5 - 36
The vertex is at (6, -31)
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 - 28x + 200
y = x2 - 28x + ____ + 200
y = x2 - 28x + 196 + 200 - 196
y = (x - 14)2 + 4
The vertex is at
(14, 4)
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 – 0.75x - 1
y = x2 – 0.75x + ____ + - 1
y = x2 – 0.75x + .140625 - 1 - .140625
y = (x – 0.375)2 – 1.140625
The vertex is at
(0.375, -1.140625)
9
Change to vertex form:
y = x2 + 4x + 10
y = x2 + 4x + ___ + 10
y = x2 + 4x + 4 + 10 - 4
y = (x + 2)2 + 6
10
Change to vertex form:
y = x2 + 19x - 1
y = x2 + 19x + ___ - 1
y = x2 + 19x + 90.25 - 1 – 90.25
y = (x + 9.5)2 - 91.25
11
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.
12
Change to vertex form:
y = 2x2 + 4x + 10
y = 2(x2 + 2x + ___) + 10 - ___
y = 2(x2 + 2x + 1) + 10 - 2
y = 2(x + 1)2 + 8
Vertex at (-1, 8)
13
Change to vertex form:
y = 3x2 + 12x + 22
y = 3(x2 + 4x + ___) + 22 - ___
y = 3(x2 + 4x + 4) + 22 - 12
y = 3(x + 2)2 + 10
Vertex at (-2, 10)
14
Change to vertex form:
y = 6x2 - 48x + 65
15
Change to vertex form:
y = 7x2 - 98x + 400
16
Change to vertex form:
y = 12x2 - 60x + 312
17
Change to vertex form:
y = -5x2 + 20x - 32
y = -5(x2 - 4x + ___) - 32 - ___
y = -5(x2 - 4x + 4) - 32 + 20
y = -5(x - 2)2 - 12
Vertex at (2, -12)
18
Change to vertex form:
y = -6x2 + 72x - 53
y = -6(x2 - 12x + ___) - 53 - ___
y = -6(x2 - 12x + 36) - 53 + 216
y = -6(x - 6)2 + 163
Vertex at (6, 163)
19
Methods of Locating the Vertex of a Parabola:
If the quadratic is in vertex form:
𝑦 =𝑎 𝑥−ℎ
2
+𝑘
The vertex is @ (h, k):
If the quadratic is in factored form: The x value of the
vertex is halfway
𝑦 = 𝑎 𝑥 − __ 𝑥 − __
between the roots.
Plug in & solve to
find the y value.
If the quadratic is in standard form: Complete the
square to change to
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
vertex form.
20
Change to vertex form:
2
y  5x  3x  2
 2 3

y  5 x  x  ___  2  ___


5
 2 3
9 
45
y  5 x  x 
 2 

5
100 
100
2

3  245
y  5 x 
 

10  100
Vertex at (-0.3, -2.45)
21
Change to vertex form:
2
y  7 x  9 x  25
 2 9

y   7 x  x  ___  25  ___


7
 2 9
81 
81
y  7 x  x 
 25 

7
196 
28
2

9  619
y  7 x 
 

14 
28
 9 619 
,


 14 28 
22
Change to vertex form:
y  5 x  8 x
2
23
Change to vertex form:
y
1
x  2x  3
2
2
24
Solve by completing the square.
x
2
 4x  5
x  4x  5
2
x  4 x  __  5  __
x  2 
2
x  2 
2
9

9
2
x  4x  4  5  4
x  2  3
2
x  2 
2
x  3  2
9
x  5, 1
25
Solve by completing the square.
2 x  5  12 x
2
2 x  12 x  5
x  3
2

2
2
2  x  6 x  __   5  __
2
2  x  6 x  9   5  18
2
2  x  3   23
23
2
x  3 
2
x  3
2
23

x3 
2
x  3
23
2
23
2
23
2
x  6 . 391 ,  0 26. 391
Example: Solve by completing the square:
x2 + 6x – 8 = 0
x2 + 6x - 8 = 0
x2 + 6x = 8
x2 + 6x + ___= 8 + ___
x2 + 6x + 9 = 8 + 9
(x+3)2 = 17
x  3   17
x   3  17
27
Solve by completing the square:
0  ax
ax
2
2
 bx  c
 bx   c
 2 b

a  x  x  __    c  __
a


 2 b
b
a  x  x 
2
a
4a

2
2

b
  c 

4a

2
2
b 
b

a x 
c
 
2a 
4a

28
Solve by completing the square:
2
2
b 
b

a x 
c
 
2a 
4a

2
2
b 
b
c


x
 
2
2a 
a
4a

b 

x

2a 

2

b
2
4a
2
x
b
 
2a
x 
b
2a
c x   b 

2a
a
2
b
4a
b

2
2
4a

a
2
4a
b
2
2

c

c
a
4 ac
4a
29
2
Solve by completing the square:
b
x  

b
2
2

4 ac
2a
4a
b
b  4 ac
x 
2

2a
x  
b
2a
4a
2
4a
2
b  4 ac
2

2a
x 
b
b  4 ac
2
2a
This is called the
Quadratic Formula.
You must memorize it!!!
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