Factoring Worksheet With Examples Page 37
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50Perfect Squares and
8-6
Factoring
Main Ideas
The senior class has decided to build an
8 ft
•
Factor perfect square
outdoor pavilion. It will have an 8-foot by
trinomials.
x x
8-foot portrayal of the school’s mascot in the
•
Solve equations
center. The class is selling bricks with students’
involving perfect
squares.
names on them to finance the project. If they
x x
x x
8 ft
sell enough bricks to cover 80 square feet and
New Vocabulary
want to arrange the bricks around the art, how
x x
wide should the border of bricks be?
perfect square trinomials
To solve this problem, you need to solve the
2
equation (8 + 2x )
= 144.
Factor Perfect Square Trinomials
Numbers like 16, 49, and 144 are
perfect squares, since each can be expressed as the square of an integer.
2
2
2
16 = 4 · 4 or 4
49 = 7 · 7 or 7
144 = 12 · 12 or 1 2
2
2
2
Products of the form (a + b )
and (a - b )
, such as (8 + 2x )
, are also
perfect squares. Recall that these are special products that follow specific
patterns.
2
2
(a + b )
= (a + b)(a + b)
(a - b )
= (a - b)(a - b)
2
2
2
2
= a
+ ab + ab + b
= a
- ab - ab + b
2
2
2
2
= a
+ 2ab + b
= a
- 2ab + b
These patterns can help you factor perfect square trinomials , which are
trinomials that are the squares of binomials.
Squaring a Binomial
Factoring a Perfect Square
2
2
2
2
2
2
(x + 7 )
= x
+ 2(x)(7) + 7
x
+ 14x + 49 = x
+ 2(x)(7) + 7
2
2
= x
+ 14x + 49
= (x + 7 )
2
2
2
2
2
2
(3x - 4 )
= (3x )
- 2(3x)(4) + 4
9 x
- 24x + 16 = (3x )
- 2(3x)(4) + 4
2
2
= 9 x
- 24x + 16
= (3x - 4 )
For a trinomial to be factorable as a perfect square, three conditions must
be satisfied as illustrated in the example below.
2
+ 20x + 25
4 x
The fi rst term must
The last term must
be a perfect square.
be a perfect square.
2
2
2
The middle term must be twice
4x
= (2x)
25 = 5
the product of the square roots
of the fi rst and last terms.
2(2x)(5) = 20x
454
Chapter 8 Factoring
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