Factorization by the Difference of squares

Definition

A method of factorizing an expression in the form of difference of two squares is called the factorization of an expression by the difference of squares.

Introduction

The square of a number can be subtracted easily from the square of another number, but it is not possible to subtract an expression in square form from another expression in square form when the expressions are unlike. In this case, the difference between the expressions is written as an expression simply.

However, the difference between the terms in square form in an expression can be factorized to convert it as a product of two or more factors and it is called the factorization by the difference of squares.

The following expressions are the best examples for the expressions in the difference of two squares form.

1. $x^2-16$
2. $441-81y^2$
3. $64x^2-121y^2$

Let’s learn how to factorize an expression in the form of the difference of two squares.

Formulas

The difference of squares rule is used as a formula to factorize the expressions in the form of the difference of squares, and it is written popularly in the following two mathematical forms.

1. $a^2-b^2$ $\,=\,$ $(a+b)(a-b)$
2. $x^2-y^2$ $\,=\,$ $(x+y)(x-y)$

Method

There are two simple steps to factorize any expression in the form of the difference of squares.

1. Write each term of the expression in square form.
2. Use the difference of two squares formula to factorize the expression as a product of two factors.

Examples

Let’s learn how to factorise an expression by the difference of squares from the below example.

Factorize $16x^2-49y^2$

Write every term in square form

Write each term in square form to apply the difference of squares rule in factor form.

$=\,\,\,$ $4^2x^2-7^2y^2$

$=\,\,\,$ $(4x)^2-(7y)^2$

Factorize by the sum of squares rule

Take $a = 4x$ and $b = 7y$. Now, express the difference of the squares in factor form.

$=\,\,\,$ $(4x+7y)(4x-7y)$

Worksheet

$(1).\,\,$ $16a^4-81b^4$

$(2).\,\,$ $8xy^2-18x^3$

$(3).\,\,$ $(x-2y)^2-z^2$

List of the math questions with solutions to learn how to factorize the difference of squares by using the difference of squares formula.

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