Factor a Difference of Squares

So far we’ve learned three factoring techniques. The first one, common factoring, is a technique that can be used for any polynomial. The other two were specific for quadratics, namely trial-and-error and decomposition.

The whole purpose behind factoring any quadratic – if you haven’t discovered already – is convert it in such a format that enables you to solve for the x’s (the roots). Otherwise, if it’s in general form, you can’t easily do that.

You’ll find out here that some quadratics whose b-term is missing and whose a and c terms are being subtracted, can be factored by another technique known the difference of squares. Examples of quadratics that fit this pattern look like this:

  1. y = x² – 100   (easy)
  2. y = 98a² – 450b²   (medium)
  3. y = (3x + 8)² – (x – 2)²   (hard)

Also, as the name of the technique implies, both terms in the quadratic need to be perfect squares (squarootable).

Take equation (1) as an example. Taking the square root of x² yields x. In other words, √x² = x. Similarly, √100 = 10. In the second equation, after common factoring 2 from both terms, you get:

2(49a² – 225b²)

Notice now that √49a² = 7a and √225b² = 15b. Both terms within the parentheses are perfect squares; hence, this technique can be used.

In equation (3), the square root of (3x + 8)² is 3x + 8 and the square root of (x – 2)² is x – 2; therefore it fits the same mold. A summary of what’s required is written below.

Conditions required to factor a difference of squares:

  • b term needs to be missing.
  • The two remaining terms are being subtracted.
  • The two remaining terms need to be a perfect square.

Let’s what a few examples: