Skill: Factor differences of squares

A difference of squares has the form

$$a^2-b^2$$

It factors using the pattern

$$a^2-b^2=(a-b)(a+b).$$

The expression must have:

• subtraction between two terms, and
• each term must be a perfect square.

Example: Factor $(25x^2-16)$

Step 1: Recognize the two squares.

Both terms are perfect squares:

$$25x^2=(5x)^2$$

and

$$16=4^2.$$

So rewrite the expression as

$$25x^2-16=(5x)^2-4^2.$$

Step 2: Apply the difference-of-squares pattern.

Here, $(a=5x)$ and $(b=4)$. Therefore,

$$(5x)^2-4^2=(5x-4)(5x+4).$$

So the factored form is

$$\boxed{(5x-4)(5x+4)}.$$

Step 3: Check by multiplying.

Use the distributive property:

$$(5x-4)(5x+4) =25x^2+20x-20x-16 =25x^2-16.$$

The middle terms cancel, confirming the answer.

Remember: this pattern applies to a difference, or subtraction, of squares. An expression such as $(x^2+9)$ is a sum of squares and does not factor using this pattern over the real numbers.