# $(a-b)^2$ Formula

## Formula

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

### Introduction

Let $a$ and $b$ represent two terms in algebraic form. The difference of them is equal to $a-b$, which is a binomial and the square of the binomial $a-b$ is expressed as $(a-b)^2$ in mathematical form. It is also known as the square of the difference of the two terms.

The $a-b$ whole square is used as a formula to expand it as an algebraic expression $a^2-2ab+b^2$ in mathematics.

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

##### Usage

The square of the difference of two terms formula is used in mathematics in two different cases.

1. It is used to expand the square of difference of two terms or a binomial.
2. If a mathematical expression is in the form of $a^2+b^2-2ab$, then it is simply written as $(a-b)^2$.

#### Examples

$(1) \,\,\,$ Expand $(2x-5y)^2$

Take $a = 2x$ and $b = 5y$, and replace them in the expansion of the formula for calculating the value in mathematical form.

$\implies$ $(2x-5y)^2$ $\,=\,$ $(2x)^2+(5y)^2-2(2x)(5y)$

$\implies$ $(2x-5y)^2$ $\,=\,$ $4x^2+25y^2-2 \times 2x \times 5y$

$\,\,\, \therefore \,\,\,\,\,\,$ $(2x-5y)^2$ $\,=\,$ $4x^2+25y^2-20xy$

$(2) \,\,\,$ Simplify $9m^2+16n^2-24mn$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-24mn$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times 3 \times 4 \times m \times n$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times 3 \times m \times 4 \times n$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times (3m) \times (4n)$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2(3m)(4n)$

Take $a = 3m$ and $b = 4n$. Now, simplify the algebraic expression by the $(a-b)^2$ algebraic identity

$\,\,\, \therefore \,\,\,\,\,\,$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m-4n)^2$

### Proofs

The $a$ minus $b$ whole squared identity can be derived mathematically in two different methods.

1. Learn how to derive the expansion of the $a-b$ whole square formula by the multiplication in algebraic approach.

2. Learn how to derive the expansion of the $a-b$ whole squared identity by the areas of geometric shapes.

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