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

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.

- The square of difference of the terms formula.
- The square of a binomial identity.
- The special binomial product rule.

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

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.

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

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.

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

$(1) \,\,\,$ 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

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

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

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

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

Latest Math Topics

Nov 03, 2022

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

Oct 24, 2022

Sep 30, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved