# How to Factor a Difference of Squares?

To factor a difference of squares, we need to start by applying a square root to both terms of the expression given. Then, we write the algebraic expression as a product of the sum of the terms and a difference of the terms. The difference of squares theorem tells us that if we have an expression of the form a²-b², this is equivalent to (a+b)(a–b).

In this article, we will look at the difference of squares in more detail. We will look at how to factor the difference of squares by using a formula, and we will look at worked-out examples to understand the concepts.

##### ALGEBRA

Relevant for

Learning to solve exercises with difference of squares.

See process

##### ALGEBRA

Relevant for

Learning to solve exercises with difference of squares.

See process

## Definition of difference of squares

The difference of two squares is a theorem that tells us if a quadratic equation can be written as a product of two binomials, where one shows the difference of the square roots and the other shows the sum of the square roots.

A difference of squares is something that looks like $latex {{x}^2} – 4$. This is because $latex {{2} ^ 2} = 4$, so we actually have $latex {{x} ^ 2}- {{2} ^ 2}$, which is a difference of squares.

## Difference of squares formula

The difference of squares formula is an algebraic expression that is used to express the difference between two squared values. A difference of squares is expressed in the form:

$latex {{a}^2} – {{b}^2}$

where the first and last terms are perfect squares.

Factoring the difference of squares, we have:

$latex {{a}^{2}}-{{b}^{2}}=(a+b)(a-b)$

This is true because $$(a+b)(a-b)={{a}^{2}}+ab-ab-{{b}^{2}}={{a}^{2}}-{{b}^{2}}$$

## How to factor a difference of squares

The following are the steps required to factor a difference of squares:

Step 1: Determine if the terms have something in common, called the greatest common factor. If so, factor out that common factor, and don’t forget to include it in your final answer.

Step 2: All difference of squares problems can be factored as follows: $latex {{a}^2}-{{b}^2} = (a + b) (a-b)$. Therefore, all we have to do is to find the square numbers that will produce the desired results.

Step 3: Determine if the remaining factors can be factored further.

## Difference of squares – Examples with answers

The following examples follow the process outlined above to factor the difference of squares:

### EXAMPLE 1

Factor $latex {{x}^{2}}-9$.

Step 1: Determine if the terms have anything in common: they have nothing in common.

Step 2: To factor the problem in the form $latex (a+b)(a-b)$, we need to determine the value that we have to square to get $latex {{x}^{2}}$ and the value that we have to square to obtain $latex 9$. In this case, we have x and 3 because $latex (x)(x)={{x}^{2}}$ and $latex (3)(3)=9$.

$latex (x+3)(x-3)$

Step 3: Determine whether the remaining factors can be factored: in this case no.

### EXAMPLE 2

Factor $latex 4{{x}^{2}}-49$.

Step 1: Determine if the terms have anything in common: they have nothing in common.

Step 2: To factor the problem in the form $latex (a+b)(a-b)$, we need to determine the value that we have to square to get $latex 4{{x}^{2}}$ and the value that we have to square to obtain $latex 49$. In this case, we have 2x and 7 because $latex (2x)(2x)=4{{x}^{2}}$ and $latex (7)(7)=49$.

$latex (2x+7)(2x-7)$

Step 3: Determine whether the remaining factors can be factored: in this case no.

### EXAMPLE 3

Factor $latex 18{{x}^{2}}-98$.

Step 1: Determine if the terms have anything in common: they have 2 common.

$latex 2(9{{x}^2}-49)$

Step 2: To factor the problem in the form $latex (a+b)(a-b)$, we need to determine the value that we have to square to get $latex 9{{x}^{2}}$ and the value that we have to square to obtain $latex 49$. In this case, we have 3x and 7 because $latex (3x)(3x)=9{{x}^{2}}$ and $latex (7)(7)=49$.

$latex 2(3x+7)(3x-7)$

Step 3: Determine whether the remaining factors can be factored: in this case no.

### EXAMPLE 4

Factor $latex 4{{x}^{2}}-64$.

Step 1: Determine if the terms have anything in common: they have 4 common.

$latex 4({{x}^2}-16)$

Step 2: To factor the problem in the form $latex (a+b)(a-b)$, we need to determine the value that we have to square to get $latex {{x}^{2}}$ and the value that we have to square to obtain $latex 16$. In this case, we have x and 4 because $latex (x)(x)={{x}^{2}}$ and $latex (4)(4)=16$.

$latex 4(x+4)(x-4)$

Step 3: Determine whether the remaining factors can be factored: in this case no.

### EXAMPLE 5

Factor $latex 16{{x}^{4}}-1$.

Step 1: Determine if the terms have anything in common: they have nothing in common.

Step 2: To factor the problem in the form $latex (a+b)(a-b)$, we need to determine the value that we have to square to get $latex 16{{x}^{4}}$ and the value that we have to square to obtain $latex 1$. In this case, we have $latex 4{{x}^2}$ and 1 because $latex (4{{x}^2})(4{{x}^2})=16{{x}^{4}}$ and $latex (1)(1)=1$.

$latex (4{{x}^2}+1)(4{{x}^2}-1)$

Step 3: Determine whether the remaining factors can be factored: in this case one of the factors is a difference of squares, so we can factor it: we need to determine the value we have to square to get $latex 4{{x}^2}$ and the value we have to square to obtain 1. In this case, we have $latex 2x$ and 1 since $latex (2x) (2x) = 4 {{x} ^2}$ and $latex (1)(1) = 1$.

$latex (4{{x}^{2}}+1)(2x+1)(2x-1)$

## Difference of squares – Practice problems

#### Find the expression that is equal to $latex 18x^4-32$.  