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

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.

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

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

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

$(1) \,\,\,$ 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 best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
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.