Factoring the Difference of Two Squares
(a² - b²)

To factor the difference of two squares, you must first know what a "squared" number means. A number multiplied by itself results in a squared number. The first ten perfect squares are listed below.

1² = 1 x 1 = 1
2² = 2 x 2 = 4
3² = 3 x 3 = 9
4² = 16
5² =25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100

It is important to memorize the first ten perfect squares. Otherwise, when you need to factor a polynomial, you won't recognize what technique to use. Until you've memorized this list, refer back to it for each problem.

Perfect square polynomials occur when you foil two binomials and the middle terms cancel eachother out.
For example, if you use FOIL to multiply (x - 4)(x + 4) you get x² + 4x - 4x - 16 which equals x² - 16.

To factor the difference of two squares, they give you the answer, x² - 16, and you have to determine what two binomials FOILed to make that answer. In this case, (x + 4)(x - 4).

The Formula to Factor the Difference of Two Squares
x² - a² =
(x + a)(x - a)

Example:
Factor x² - 4
Step 1: Rewrite 4 as some number squared. Refer to the list above until you have it memorized.
x² - 4 = x² - (2)²
Using the formula above, with a = 2, x² - a² = (x + a)(x - a) this expression becomes
(x + 2)(x - 2) *Answer

Example:
Factor x² - 25
Step 1: Rewrite 25 as some number squared. Refer to the list above until you have it memorized.
x² - 25 = x² - (5)²
with a = 5. Using the formula, this expression becomes
(x + 5)(x - 5) *Answer

Example:
Factor x² - 81
Step 1: Rewrite 81 as some number squared. Refer to the list above until you have it memorized.
x² - 81 = x² - (9)²
with a = 9. Using the formula, this expression becomes
(x + 9)(x - 9) *Answer

Example:
Factor x² - 49
x² - 49 = x² - (7)²
Using the formula, this expression becomes
(x + 7)(x - 7) *Answer

Example:
Factor x² - 121
x² - 121 = x² - (11)²
Using the formula, this expression becomes
(x + 11)(x - 11) *Answer

The Formula to Factor the Difference of Two Squares
ax² - b² =
(ax - b)(ax + b)

Example:
Factor 25x² - 16

Example:
Factor 49x² - 9

To check your answer, you can FOIL the binomials above and verify that your answer results in the original problem that you started with.

Remember: FOIL-ing and Factoring
are "Inverse Operations"
Each one undoes the other.