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.

## What is the difference of squares?

The difference of two squares is a theorem that tells us that a expression of the form $latex a^2-b^2$ 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.

## Formula for the difference of squares

The formula for the difference of squares 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 given that $$(a+b)(a-b)=a^2+ab-ab-b^2=a^2-b^2$$

## How to factor a difference of squares?

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

#### 1. Factor the initial expression if possible.

Determine if the terms have a common factor. If so, factor out that common factor, and don’t forget to include it in the final answer. For example, $latex 2x^2-32=2(x^2-16)$.

#### 2. Apply the difference of squares formula.

The formula for the difference of squares is $latex a^2-b^2=(a+b)(a-b)$. For example, $latex 2(x^2-16)$ is equal to $latex 2(x+4)(x-4)$.

#### 3. Factor and simplify the final result.

Determine whether the remaining factors can be further factored or simplified.

## Solved examples of the difference of squares

### EXAMPLE 1

Factor $latex {{x}^{2}}-9$.

##### Solution

**Step 1:** In this case, the expression doesn’t need to be factored.

**Step 2:** To factor the problem in the form $latex (a+b)(a-b)$ we need to determine the value which we have to square to get $latex {{x}^{2}}$ and the value which we have to square to get 9.

In this case, we have *x* and 3, since $latex (x)(x)={{x}^{2}}$ and $latex (3)(3)=9$.

$latex {{x}^{2}}-9$

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

**Step 3:** The expression is now in its simplest form.

### EXAMPLE 2

Use the difference of squares to factor $latex 4x^2-49$.

##### Solution

**Step 1:** The terms don’t have a common factor.

**Step 2:** To factor the problem in the form $latex (a+b)(a-b)$ we need to determine the value we need to square to get $latex 4x^2$ and the value we need to square to get 49.

In this case, we have 2*x* and 7 because $latex (2x)(2x)=4x^2$ and $latex (7)(7)=49$.

$latex 4x^2-49$

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

**Step 3:** The expression is now in its simplest form.

### EXAMPLE 3

Factor the expression $latex 18x^2-98$.

##### Solution

**Step 1:** The terms have a common factor:

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

**Step 2:** We have to determine the value we need to square to get $latex 9x^2$ and the value we need to square to get 49.

In this case, we have 3*x* and 7 because $latex (3x)(3x)=3x^2$ and $latex (7)(7)=49$.

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

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

**Step 3:** The expression is now in its simplest form.

### EXAMPLE 4

Use the difference of squares in the expression $latex 4x^2-64$.

##### Solution

**Step 1:** The terms have a common factor.

$latex 4(x^2-16)$

**Step 2:** We have to find the value we need to square to get $latex x^2$ and the value we need to square to get 16.

Therefore, we have *x* and 4 because $latex (x)(x)=x^2$ and $latex (4)(4)=16$.

$latex 4(x^2-16)$

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

**Step 3:** The result can no longer be simplified.

### EXAMPLE 5

Factor the expression $latex 16x^4-1$.

##### Solution

**Step 1:** We have no common factors.

**Step 2:** We have to find the value we need to square to get $latex 16x^4$ and the value we need to square to get 1.

In this case, we have $latex 4x^2$ and 1 because $latex (4x^2)(4x^2)=16x^4$ and $latex (1)(1)=1$.

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

**Step 3:** One of the factors is a difference of squares, so we can factor it: we have to determine the value we need to square to get $latex 4x^2$ and the value we need to square to get 1.

In this case, we have $latex 2x$ and 1 because $latex (2x)(2x)=4x^2$ and $latex (1)(1)=1$.

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

If you want to explore more examples of difference of squares, you can visit our article: Difference of Squares – Examples and Practice Problems.

## Difference of squares – Practice

#### Factor the expression $latex 4x^2-36$ using the difference of squares.

Write the answer in the input box.

## See also

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