# Solving Quadratic Equations by Completing the Square

```Solving Quadratic
Equations by
Completing the Square
Perfect Square Trinomials
Examples
 x2 + 6x + 9
 x2 - 10x + 25
 x2 + 12x + 36

Creating a Perfect
Square Trinomial
In the following perfect square
trinomial, the constant term is
missing.
X2 + 14x + ____
 Find the constant term by
squaring half the coefficient of
the linear term.
 (14/2)2
X2 + 14x + 49

Perfect Square Trinomials
Create perfect
square trinomials.
 x2 + 20x + ___
 x2 - 4x + ___
 x2 + 5x + ___

100
4
25/4
by Completing the Square
Solve the following
equation by
completing the
square:
Step 1: Move
linear term to left
side of the
equation
x  8 x  20  0
2
x  8 x  20
2
by Completing the Square
Step 2:
Find the term
that completes the square
on the left side of the
to both sides.
x  8x 
2
=20 +
1
 (8)  4 then square it, 42  16
2
x  8 x  16  20  16
2
by Completing the Square
Step 3:
Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
x  8 x  16  20  16
2
( x  4)( x  4)  36
( x  4)  36
2
by Completing the Square
Step 4:
Take the
square
root of
each side
( x  4)  36
2
( x  4)  6
by Completing the Square
Step 5: Set
up the two
possibilities
and solve
x  4  6
x  4  6 and x  4  6
x  10 and x=2
Completing the Square-Example #2
Solve the following
equation by completing
the square:
term, and linear term to
left side of the equation,
the constant to the right
side of the equation.
2 x 2  7 x  12  0
2 x  7 x  12
2
by Completing the Square
Step 2:
Find the term
that completes the square
on the left side of the
to both sides.
must be equal to 1 before
you complete the square, so
you must divide all terms
coefficient first.
2 x2  7 x 
=-12 +
2 x2 7 x
12


 
2
2
2
2 7
x  x   6 
2
1 7
7
 ( )  then square it,
2 2
4
7
49
49
x  x
 6 
2
16
16
2
2
49
7
  
 4  16
by Completing the Square
Step 3:
Factor
the perfect
square trinomial
on the left side
of the equation.
Simplify the
right side of the
equation.
7
49
49
x  x
 6 
2
16
16
2
2
7
96 49

x   
4
16 16

2
7
47

x  
4
16

Completing the Square
Step 4:
Take the
square
root of
each side
7 2
47
(x  ) 
4
16
7
47
(x  )  
4
4
7 i 47
x 
4
4
7  i 47
x
4
Completing the Square
Try the following examples. Do your work on your paper and then check
1. x  2 x  63  0
2
2. x  8 x  84  0
2
3. x  5 x  24  0
2
4. x  7 x  13  0
2
5. 3 x 2 5 x  6  0
1.  9, 7 
2.(6, 14)
3.  3,8 
 7  i 3 
4. 

2


 5  i 47 
5. 

6


```