Another common and useful method of factoring is called the difference of squares method. Whenever you have two perfect square factors being subtracted, you have a difference of squares. The first step in factoring using this method is identification:

Identifying a Perfect Square Term

A perfect square term can be factored into two identical whole-number terms. For example, 4 is a perfect square because it can be broken down into two identical whole-number terms: 2*2.

4x2 is a perfect square because the square root of 4x² is +/-2x

9x³ is not a perfect square because the square root of 9x³ is 3x root x, which contains a radical expression.

Factoring the Difference of Squares

Now that you can identify perfect squares, let’s look at an example. Here is an expression:

x² - y²

As you can see, this is a difference of squares because we are calculating the difference between y² and x², two perfect square terms. Ok, so what can we do with this? We can factor by taking the square root of the first term minus the square root of the second term, and multiplying that quantity by the square root of the first term plus the square root of the second term. Let’s see this in action:

x² - y² = (x-y)(x+y)

That’s all there is to it! Now let’s use this formula as we try another example:

9a⁶ – 16b²

Ok, now this one looks a little more complicated. First, we need to decide if this is a difference of squares.

9a⁶ = 3a³*3a³ - Meets criteria for a perfect square

16b² = 4b*4b - Meets criteria for a perfect square

Now that we have identified this expression as a difference of squares, let’s factor it using the form we explained above:

x² - y² = (x-y)(x+y)

In our example x² is now 9a⁶ and y² is now 16b²

9a⁶ – 16b² = (3a³-4b)(3a³+4b)

And we’re done!