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Completing the Square
An easier way to factor the trial-and-error
style equations to solve.
Remember the Binomial Square?
Use Foil to multiply out:
( x  4)
( x  4)( x  4)
Or use the shortcut:
2
x  4 x  4 x  16
2
x  8 x  16
2
( x  4)
2
Scratch work!
2 1 4  8
x  8 x  16
2
Remember the Binomial Square?
Check to see if it is a perfect square:
First, Check if the first and last numbers are perfect
squares.
Second, take the square roots of the first and last terms,
multiply them and then multiply them by 2. If you get the
middle number, you’ve got a perfect square!
(1*4)*2 = (4)2 = 8
1x
2
 8 x  16
( x  4)
2
Remember the Binomial Square?
Check to see if it is a perfect square:
First, Check if the first and last numbers are perfect
squares.
Second, take the square roots of the first and last terms,
multiply them and then multiply them by 2. If you get the
middle number, you’ve got a perfect square!
(1*2)*2 = (2)2 = 4
1x
2
 2x  4
Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
______________________
_______________________,
remembering the “” on the righthand side. Simplify as necessary.
4x  2x  5  0
2
5 5
4x  2x  5
2
4
4
4
1
5
x  x
2
4
2
2
2
1
 1 1  1
      
 2 2   4  16
Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
Now factor the right side.
______________________
Take the square root of both sides
____________________________,
remembering the “” on the righthand side. Simplify as necessary.
1
1 5 1
x  x  
2
16 4 16
2
1
1 21
x  x 
2
16 16
2
2
1
21

x  
4  16

2
1
21

x  
4
16

Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
Now factor the right side.
______________________
Take the square root of both sides
____________________________,
remembering the “” on the righthand side. Simplify as necessary.
1
x   1.146
4
Solve for x:
1
x   1.146
4
1
x   1.146 x  1  1.146
4
4
 .896
 1.396
Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
______________________
_______________________,
remembering the “” on the righthand side. Simplify as necessary.
10 x  20 x  71  9
2
 71  71
10 x  20 x  80
2
10
10
10
x  2x  8
2
2
 1
2


2


1
1


 2
Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
Now factor the right side.
______________________
Take the square root of both sides
____________________________,
remembering the “” on the righthand side. Simplify as necessary.
x  2x 1  8 1
2
x  2x 1  9
2
x  1
2
x  1
2
9
 9
Now to Completing the Square
This is the original problem.
constant
Move the __________over
to the
other side.
the
Divide through by ______
Leading coefficient
____________________.
half of the b-value and
Take _________________
square it. Add this square to both
sides of the equation.
Now factor the right side.
______________________
Take the square root of both sides
____________________________,
remembering the “” on the righthand side. Simplify as necessary.
x 1  3
Solve for x:
x  1 3
x  1  3
2
x  1  3
 4