# $(a-b)^2$ Identity

## Formula

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

### Introduction

Let the literals $a$ and $b$ represent two terms in algebraic form. The subtraction of $b$ from $a$ is written as $a-b$. It is basically an algebraic expression and also a binomial.

The square of this expression is written as $(a-b)^2$ in mathematical form and it is expanded as $a^2-2ab+b^2$ mathematically.

In mathematics, this algebraic identity is used as a formula and it is called in the following three ways.

1. The square of difference of the terms formula.
2. The square of a binomial identity.
3. The special binomial product rule.

#### Usage

The square of difference of terms is used as a formula in mathematics in two cases.

##### Expansion

The square of difference of the terms is expanded as the subtraction of two times product of two terms from the sum of the squares of the terms.

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

##### Simplification

The subtraction of two times product of two terms from the sum of the squares of the terms is simplified as the square of difference of the terms.

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

##### 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.

##### Algebraic method

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

##### Geometric method

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

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