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Factoring

Difference of Two Squares

There are binomials that can be more easily factored if the pattern is recognized. One of these binomials is the difference of squares:

a2 - b2 = (a + b)(a - b)

Notice that there is no middle term in the binomial. When (a + b) and (a - b) are multiplied the new expression is a2 + ab - ab - b2, since the two middle terms cancel the remaining expression is a2 - b2. This pattern only works when the two terms are subtracted.

Example 1 Factor x2 -16

Step 1. Determine if the first and last terms are perfect squares.

The first term is 1x2, which is (1x)2 - yes.

The last term is 16, which is (4)2 - yes.

Step 2. Since the two terms are also being subtracted, substitute values for a and b into the Difference of Squares formula.

x2 - 16 = (1x)2 - (4)2 = (x + 4)(x - 4)

Example 2 Factor 9x2 - 4y2

Step 1. Determine if the first and last terms are perfect squares.

The first term is 9x2, which is (3x)2 - yes.

The last term is 4y2, which is (2y)2 - yes.

Step 2. Since the two terms are also being subtracted, substitute values for a and b into the formula.

9x2 - 4y2 = (3x + 2y)(3x - 2y)