# What Is the Difference of Squares?

The difference of squares is a special type of quadratic equation:

# How to Factor a Difference of Squares

A difference of squares always factors to a simple expression:

Expand the brackets using the FOIL method to show this is the case (see Note).

# A Real Example of Factoring a Difference of Squares

Question: Factor and solve the difference of squares below:

Factoring

Step 1
Check that the equation is a difference of squares.
Is the constant term, 9, a square number? Yes.

9 = 32

Step 2
Find the values of a and b.
Compare a2 - b2 with x2 - 9.

a = x, b = 3

Step 3
Factor as (a + b)(a - b).

Solution

Equate the quadratic equation to 0 and solve for x. Each bracket in the factored equation can be equated to 0 in turn and solved for x

Step 4
Equate the first bracket to 0, and use algebra to find the first root:

Step 5
Equate the second bracket to 0, and use algebra to find the second root:

The solution to the quadratic equation x2 - 9 = 0 is x = -3 or x = 3.

# Another Real Example of How to Factor a Difference of Squares

The slider below shows another real example of how to factor a difference of squares.
show

##### Note
WHAT'S IN A NAME?

A difference of squares is so called because it is a difference (the result of subtraction) of two square numbers.

# WHY THIS FACTORING WORKS

Factoring is the opposite of expanding two brackets using the FOIL method.

Let's go backwards. Start with the difference of squares factored into two brackets:

Expand the brackets using the FOIL method:

...which is a difference of squares!