Difference of squares Rule

The difference of the squares of any two quantities is equal to the product of their sum and difference, is called the difference of squares rule.

Introduction

Let the literals $a$ and $b$ denote two variables.

The addition of them forms a binomial $a+b$.
The subtraction of them forms another binomial $a-b$.
The difference of their squares also forms another binomial $a^2-b^2$.

According to a mathematical property, the difference of the squares of any two quantities is exactly equal to the product of their sum and difference. It can be written in mathematical form as follows.

$\implies$ $a^2-b^2$ $\,=\,$ $(a+b) \times (a-b)$

$\,\,\,\therefore\,\,\,\,\,\,$ $a^2-b^2$ $\,=\,$ $(a+b)(a-b)$

It is an algebraic identity and it is called the difference of squares law.

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Other form

The difference of squares identity is also expressed in terms of variables $x$ and $y$ alternatively.

$x^2-y^2$ $\,=\,$ $(x+y)(x-y)$

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Uses

The difference of squares algebraic identity is mainly used as a formula in two cases in mathematics.

It is used to find the difference of squares of two quantities by calculating the product of sum and difference of the quantities.
It is also used to convert the difference of squares of two quantities into the factor form for simplifying expressions and solving the equations.

List of the understandable examples to learn how to use the difference of squares rule in mathematics.

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Verification

Take $a \,=\, 4$ and $b \,=\, 2$. Now, evaluate both sides of the expressions by substituting the values.

$(1).\,\,$ $a^2-b^2$ $\,=\,$ $4^2-2^2$ $\,=\,$ $16-4$ $\,=\,$ $12$

$(2).\,\,$ $(a+b)(a-b)$ $\,=\,$ $(4+2)(4-2)$ $\,=\,$ $6 \times 2$ $\,=\,$ $12$

Similarly, take $x \,=\, 7$ and $y \,=\, 3$. Now, calculate values of both sides of the equation.

$(1).\,\,$ $x^2-y^2$ $\,=\,$ $7^2-3^2$ $\,=\,$ $49-9$ $\,=\,$ $40$

$(2).\,\,$ $(x+y)(x-y)$ $\,=\,$ $(7+3)(7-3)$ $\,=\,$ $10 \times 4$ $\,=\,$ $40$

The above two examples have proved that the difference of squares of any two numbers is exactly equal to the product of their sum and difference. In this way, the difference of squares identity can be verified arithmetically in mathematics.