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

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

Latest Math Problems

Email subscription

Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.