# Factoring a Difference of Squares Lesson 30 1 by UdE9fJ

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• pg 1
Perfect Squares
• A number times itself gives you a perfect
square.

• Example: 1 x 1 = 1
These are all
•          2x2=4          examples of
perfect squares.
•          3x3 =9
•          4 x 4 = 16
Difference of Squares
• Difference means to subtract.
• Squares – are Perfect squares

• Therefore you should have one Perfect
Square minus another square

• When you have two binomials with the
same variables and different signs your
solutions will be a difference of squares.
Examples Of Difference of Squares
• (y + 2)(y – 2) = y2 – 4

Each term is a square

• (a + b)(a – b) = a2 – b2
Each term is a square

• (2x – y)(2x + y) = 4x2 – y2
Each term is a square
Factor a Difference of Squares
• To factor a Difference of Squares, write
each term as a square:
Example: Factor 25a2 – 16b2

SOLUTION: 25a2 – 16b2 = (5a)2 – (4b)2
= (5a – 4b)(5a + 4b)

The other
The factor is a                             factor is a
difference of                               sum of the
the two terms.                              two terms
Factor a Difference of Squares

Factor: 36x2 – 49y2

SOLUTION:
36x2 – 49y2
= (6x)2 – (7y)2
= (6x – 7y)(6x + 7y)

The other
One factor is                           factor is the
the difference                          sum of the
of the two                              two terms
terms
Factor a Difference of Squares

Factor: 144x2 – 81y2                               YOU TRY
SOLUTION:
144x2 – 81y2
=(   )2 – (   )2
=(      )(         )

The other
One factor is                           factor is the
the difference                          sum of the
of the two                              two terms
terms
Factor a Difference of Squares

Factor: 144x2 – 81y2                             SOLUTION
SOLUTION:
144x2 – 81y2
= (12x)2 – (9y)2
= (12x – 9y)(12x + 9y)

The other
One factor is                         factor is the
the difference                        sum of the
of the two                            two terms
terms
Factor a Difference of Squares

Factor: 9 – y2                                      YOU TRY
SOLUTION:
9 – y2
=(   )2 – (    )2
=(        )(        )

The other
One factor is                            factor is the
the difference                           sum of the
of the two                               two terms
terms
Factor a Difference of Squares

Factor: 9 – y2                                 SOLUTION
SOLUTION:
9 – y2
= (3)2 – (y)2
= (3 - y)(3 + y)

The other
One factor is                       factor is the
the difference                      sum of the
of the two                          two terms
terms
Factor Completely
To factor the sometimes requires that you do
more than one step process.

Just like in prime factorization where you keep
breaking down the numbers until you have all
prime factors, you do the same in factoring an
expression completely.
Factor Completely
x4 – y4 = (x2)2 – (y2)2
= (x2 – y2)(x2 + y2)
= (x)2 – (y)2(x2 + y2)
This is still
a difference
of squares      = (x – y)(x + y)(x2 + y2)
so we need
to factor
again.
Factor Completely
16x4 – 16y4 = (4x2)2 – (4y2)2
= (4x2 – 4y2)(4x2 + 4y2)
= (2x)2 – (2y)2(4x2 + 4y2)
This is still
a difference
of squares      = (2x – 2y)(2x + 2y)(4x2 + 4y2)
so we need
to factor
again.
YOU TRY
Factor Completely
81a4 – b4 = (           )2 – (   )2
=(        )(           )
=
This is still
a difference
of squares      =
so we need
to factor
again.
SOLUTION
Factor Completely
81a4 – b4 = (9a2)2 – (b2)2
= (9a2 – b2)(9a2 + b2)
= (3a)2 – (b)2(9a2 + b2)
This is still
a difference
of squares      = (3a – b)(3a + b)(9a2 + b2)
so we need
to factor
again.
Factor Completely
(x + 3)2 – y2 = (x + 3 – y)(x +3 + y)

There is no
difference
of squares
here so we
are done
Factor Completely
(x - 2)2 – (y – 6)2
= [(x – 2) + (y – 6)][(x – 2) – (y - 6)]
= (x – 2 + y – 6)(x – 2 - y + 6)
= (x + y – 8)(x – y + 4)
Class work
• Make sure you have completed Lesson 30
and check your solutions in the share
folder or blog.

• Complete Lesson 30(1) worksheet

• Reminder that solutions are in the share
folder or blog.

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