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Notice that in each of these examples, we end up with a quantity in the form A 2 − B 2 . In
example 1, we have
A 2 − B 2 = x 2 − 9
= (x + 3)(x − 3)
where we have identified A = x and B = 3. In example 2, we have
A 2 − B 2 = x 2 − 81
= (x + 9)(x − 9)
where we have identified A = x and B = 9. The result that we have developed and have used
in two examples is called the difference of two squares, and is written:
A 2 − B 2 = (A + B)(A − B)
The next common factorization that is important is called a perfect square. Notice that
The perfect square is written as:
Similarly,
For example,
(x + 5) 2 = (x + 5)(x + 5)
= x(x + 5) + 5(x + 5)
= x 2 + 5x + 5x + 25
= x 2 + 10x + 25
= x 2 + 2(5x) + 5 2
(x + a) 2 = x 2 + 2ax + a 2
(x − a) 2 = (x − a)(x − a)
= x(x − a) − a(x − a)
= x 2 − ax − ax + a 2
= x 2 − 2ax + a 2
(x − 7) 2 = (x − 7)(x − 7)
= x(x − 7) − 7(x − 7)
= x 2 − 7x − 7x + 7 2
= x 2 − 14x + 49
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