Completing the Square
Page 2
Perfect Square Trinomials:
Factor:
x  6x  9
2
 x  3  x  3 
So we can rewrite
these factors as:
This is called a perfect square trinomial because
the factors are the same.
x  3
2
This fact is going to help us during the
process of completing the square!
Completing the square method:
Steps:
1.
2.
Get all variables grouped together on
one side of the equation, and all the
constants on the other side of the
equation (if coefficient of the
squared term is not one, you must
divide everything by it)
Take half of the coefficient of the nonsquared variable term, square it, and
add it to both sides
x  4x  5  0
2
#8:
5 5
x  4x  5
2
4
4  5  ___
x  4 x  ___
2
  2 2
x  4x  4  9
2
3.
4.
5.
Factor the perfect square trinomial
and write it as a binomial squared
Square root both sides to get rid of
square from the binomial (don’t
forget, when introducing a square
root into the problem, your constant
will have a +/- in front of it
Solve the two equations for the
variable to get your roots
Page 2
x  2 
2
 x  2 2
9

9
x  2  3
Roots:
 1, 5 
x23
x 5
x  2  3
x  1
Page 2
Solve the quadratic equation (a) by factoring and (b) by completing the square:
#10: x  3 x  18  0
2
 x  6  x  3   0
x6
x  3
#10: x  3 x  18  0
2
 18
 18
 x  1 . 5 2

20 . 25
x  1 .5   4 .5
x  1 .5  4 .5
 1 .5  1 .5
x6
x  1 .5   4 .5
 1 .5  1 .5
x  3
x  3 x  18
2
2.25
2.25  18  ___
x  3 x  ___
2
  1 . 5 2
 x  1 . 5 2
Roots:
 20 . 25
 3 , 6 
Page 2
Solve the quadratic equation by completing the square, and express each root in
simplest radical from.
#16:
x  10 x  23  0
 23  23
2
x  10 x   23
2
x  10 x  ___
25   23  ___
25
2
x5 2
5 5
x  5
2
Roots:
5 
x5  2
5 5
x  5
2
  5 2
 x  5 2
2
 x  5 2

2
x5  2
2 ,5 
2

Page 2
Solve the quadratic equation by completing the square, and express each root in
simplest radical from.
x  2x  7  0
7 7
2
#22:
x  2x  7
2
x  2 x  ___
1  7  ___
1
2
x 1  2 2
1
1
x  1  2 2
x  1  2 2
1
1
x  1  2 2
  1 2
8 
42
 2 2
 x  1 2
8
 x  1 2

Roots:
8
x 1   8
x  1  2 2
 1  2
2 , 1  2 2

Page 2
Solve the quadratic equation by completing the square, and express each root in
simplest radical from.
4 x  12 x  7  0
7 7
2
#24:
4 x  12 x   7
4
4
4
7
2
x  3x 
4
2
7
9
9
x  3 x  ___    ___
4
4
4
2
 3
 
 2
2
2
3
2

x  
2
4

2
3

x  
2

x

3
2
4
2
2
3
3
2
x
2
3
2

2
2
x
3
2
2

2
2


x
3
2

2
2
Roots:
3
2
 

2 
2
Page 2
Solve the quadratic equation by completing the square, and express each root in
simplest a+bi form.
x  7  4x
 4x  4x
2
#1:
x  4x  7  0
7 7
2
x2i 3
2 2
x  2i 3
x  2  i 3
2 2
x  2i 3
x  4 x  7
2
2
4
4   7  ___
x  4 x  ___
  2 2
Roots:
 x  2 2
 3
 x  2 2

3
x2   3
x  2  i 3
2  i
3 ,2  i 3

Page 2
Solve the quadratic equation by completing the square, and express each root in
simplest a+bi form.
#7:
2x  4x  3  0
3 3
2
2 x  4 x  3
2
2
2
3
2
x  2x 
2
3
2
1
1    ___
x  2 x  ___
2
2
 1
2
 x  1
 x  1
2
2

1

2
x  1  i
x  1  i
2
1
2
2
1
1
x  1 i

2

2
x  1  i
2
2
2
2
2
2
x  1 i
2
2
2
1
2

x 1   
1
x  1 i
2
2
Roots:

2
1  i

2 

Page 2
Homework
•Page 2
#19,25 top
#5,11 bottom