Math Doubts

$(a-b)^2$ 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.

  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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