Difference of squares

A squared quantity subtracted from another squared quantity is called the difference of squares.

Introduction

In mathematics, the squares of two numbers are often involved in subtraction. The difference between squares of both numbers is called as the difference of two squares or simply difference of squares.

Example

$5$ and $3$ are two numbers and the difference of squares of them is written in mathematics as follows.

$5^2-3^2$

$\implies$ $5^2-3^2$ $\,=\,$ $25-9$

$\implies$ $5^2-3^2$ $\,=\,$ $16$

In this way, the difference of squares of any two quantities is calculated mathematically in the basic mathematics.

In advanced mathematics, the difference of two squares is expressed in general form by writing it in the form of two terms but it is not possible to find the difference of squares of them, same as the above due to the unknown quantities of the terms. However, it is most useful in simplifying the complex functions in some cases by writing it into its equivalent form.

At this time, don’t think about it much more but soon you will understand how important it is in mathematics.

$5$ and $2$ are two numbers. Now, find the difference of their squares.

$\implies$ $5^2-2^2$ $\,=\,$ $5 \times 5-2 \times 2$

$\implies$ $5^2-2^2$ $\,=\,$ $25-4$

$\,\,\,\therefore\,\,\,\,\,\,$ $5^2-2^2$ $\,=\,$ $21$

Now, find the sum of the numbers and then also calculate the difference of them.

$(1).\,\,\,$ $5+2$ $\,=\,$ $7$

$(2).\,\,\,$ $5-2$ $\,=\,$ $3$

Evaluate the product of sum and difference of the numbers.

$\implies$ $(5+2) \times (5-2)$ $\,=\,$ $7 \times 3$

$\,\,\,\therefore\,\,\,\,\,\,$ $(5+2)(5-2)$ $\,=\,$ $21$

Formula

The difference of squares property can be expressed in mathematics and it is generally written in algebraic form. It is popularly written in the following two ways.

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

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

Now, learn how to write the difference of squares property in algebraic form with both algebraic and geometric proofs.