Difference of Two Squares
(2 parts)
After completing these notes, you will be ready to do the
following assignments and take the following quiz.
Assignment:
Quiz:
WS “Factoring Special Cases”
Lesson 4.4 Quiz Part 2
# 1-22 (front only)
Friday March 2nd
Objective 1: After completing part 1, students should be able to
recognize a difference of two squares.
For a binomial to be a difference of two squares the following
must be true:
• There must be two terms, both squares.
Examples of squares are:
4 x 2 and 9 y 4
16 and y 2
25x 2 y 2 and 1
• There must be a minus sign between the two terms.
Examples with a minus are:
x2  y 2
9  a 2
2
which
can
be
written
as
a
 9

Objective 2: After completing part 2, students should be able to
factor a difference of two squares.
A  B   A  B  A  B 
2
2
When you are factoring a difference of two squares, use these rules:
1. First, check to see if you can take out a GCF
2. Second, find what multiplies by itself to make the first term and the
second term.
3. Third, fill in the signs, one should be a plus, one should be a minus.
4. Finally, check that there is nothing left to factor within the
parentheses. Sometimes you can factor another difference of two
squares.
Examples :
1. x 2  25
  A   B 
2
  x    5
2
  A  B  A  B 
2
  x  5 x  5
2
2. 4 x 2  9 y 2
  A   B 
2
  2x   3 y 
2
  A  B  A  B 
2
2
  2 x  3 y  2 x  3 y 
Examples :
3. 16a 2  49
2
  A   B 
2
  4a    7 
2
2
  A  B  A  B 
  4a  7  4a  7 
4. x 2  1
  A   B 
2
  x   1
2
2
2
  A  B  A  B 
  x  1 x  1
Examples. Make sure to take out a GCF first:
5. 25x 2  9 x 4
 x  25  9 x
2
GCF: x 2
2

Then factor the difference of two squares.
 x2  5  3x  5  3x 
6. 2 x 2  50
GCF: 2
 2  x  25 
2
Then factor the difference of two squares.
 2  x  5 x  5
7. 32 y 2  8 y 6
 8y 4  y
2
 8y 2  y
2
2
GCF: 8y 2
 Then factor the difference of two squares.
 2  y 
4
2
Examples. Make sure to factor completely:
8. x  1
4
  x 2  1 x 2  1
Stays the
same.
First, factor the difference of two squares.
Then factor the difference of two squares
that’s left in the parentheses.
Factors
again.
  x 2  1 x  1 x  1
Examples. Make sure to factor completely:
9. 16  x
12
First, factor the difference of two squares.
  4  x 6  4  x 6 
Stays the
same.
 4  x
Then factor the difference of two squares
that’s left in the parentheses.
Factors
again.
6
 2  x  2  x 
3
3
Try These:
Factor.
a. 25  x
2
b. m 6  16
c. 9a8b 4  49
d. y 2  64
e. a 3b  4ab3
f. 5  20 y 6
g. 81x 4  1
h. 16m  n
4
8
If you did not get these answers, click the green
button next to the solution to see it worked out.
Solutions:
a.
 5  x  5  x 
e. ab  a  2b  a  2b 
b.
m
f. 5 1  2 y 3 1  2 y 3 
c.
 3a 4b2  7 3a 4b2  7 
g.
9x
d.
 y  8 y  8 
h.
 4m
3
 4  m  4 
3
2
2
 1  3 x  1 3 x  1
 n 4  2m  n 2  2m  n 2 
a. 25  x
2
  A   B 
2
  5   x 
2
2
2
  A  B  A  B 
  5  x  5  x 
BACK
b. m  16
6
  A   B 
2
 m

3 2
2
  4
2
  A  B  A  B 
  m  4  m  4 
3
3
BACK
c. 9a b  49
8 4
  A   B 
2
  3a b
2
  7
4 2 2
2
  A  B  A  B 
  3a b  7  3a b  7 
4 2
4 2
BACK
d. y  64
2
  A   B 
2
  y   8
2
2
2
  A  B  A  B 
  y  8 y  8
BACK
e. a b  4ab
3
3
 ab  a  4b
2
GCF: ab
2

 ab  a  2b  a  2b 
BACK
f. 5  20y
6
 5 1  4 y
6
 5 1  2 y 3
GCF: 5

1  2 y 
3
BACK
g. 81x  1
4
  9 x 2  1 9 x 2  1
Stays the
same.
Factors
again.
  9 x 2  1 3x  1 3x 1
BACK
h. 16m  n
4
8
  4m 2  n 4  4m 2  n 4 
Stays the
same.
Factors
again.

  4m 2  n 4  2m  n 2
2
2
m

n


BACK