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How to Factor a Difference of Squares (Mathematics Lesson)

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.
Interactive Test
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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!