# Square of the Difference formula

## Formula

$(1) \,\,\,$ ${(a-b)}^2$ $\,=\,$ $a^2+b^2-2ab$

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

The square of the difference of any two quantities is equals to the twice the product of them subtracted from sum of squares of them, is called the square of the difference formula.

### Introduction

The square of the difference formula is a most useful fundamental property of mathematics. It is used to find the square of difference of any two quantities by expanding its value in terms of them. In mathematics, the square of the difference formula is expressed in the following two forms. They both are same and you can use any one of them to evaluate the square of the difference of any two quantities by its expansion.

$(1) \,\,\,$ ${(a-b)}^2$ $\,=\,$ $a^2+b^2-2ab$

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

The square of the difference formula is also called as the special product of binomials because the square of binomial $a-b$ or $x-y$ can be obtained by the product of two same binomials.

$(1) \,\,\,$ ${(a-b)}^2$ $\,=\,$ ${(a-b)}{(a-b)}$

$(2) \,\,\,$ ${(x-y)}^2$ $\,=\,$ ${(x-y)}{(x-y)}$

#### Verification

Take $a = 5$ and $b = 3$, Now, substitute them in both sides of the equation and compare them for understanding this property mathematically.

$(1). \,\,\,$ Find square of the difference of both quantities.
${(a-b)}^2$ $\,=\,$ ${(5-3)}^2$
$\implies$ ${(5-3)}^2$ $\,=\,$ $2^2$
$\implies$ ${(5-3)}^2$ $\,=\,$ $4$

$(2). \,\,\,$ Find the value of the expansion of the square of the difference formula.
$a^2+b^2-2ab$ $\,=\,$ ${(5)}^2+{(3)}^2-2 \times 5 \times 3$
$\implies$ ${(5)}^2+{(3)}^2-2 \times 5 \times 3$ $\,=\,$ $25+9-30$
$\implies$ ${(5)}^2+{(3)}^2-2 \times 5 \times 3$ $\,=\,$ $34-30$
$\implies$ ${(5)}^2+{(3)}^2-2 \times 5 \times 3$ $\,=\,$ $4$

$(3). \,\,\,$ Now, compare results of them.
$\therefore \,\,\,\,\,\,$ ${(5-3)}^2$ $\,=\,$ ${(5)}^2+{(3)}^2-2 \times 5 \times 3$ $\,=\,$ $4$

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