Square of the Binomial Property

Property

The square of sum of two quantities is equal to the sum of their squares and twice their product.

Introduction

The sum of two terms raised to the power of $2$ is called the square of sum of two terms.

Two terms are often connected by a plus sign and we see the sum of them is raised to the power of $2$ in some cases in mathematics. It is generally called as square of sum of two terms and also simply called as sum of two terms whole square.

${(2m+5n)}^2$

In this example, $2m$ and $5n$ are two terms and the sum of them is $2m+5n$. This binomial is raised to the power of two. The square of sum of the two terms $2m$ and $5n$ can be calculated by multiplying the binomial by itself.

${(2m+5n)}^2$ $\,=\,$ $(2m+5n) \times (2m+5n)$

This equation states that the square of sum of two terms is equal to the product of the binomials. So, it can be evaluated by the multiplication of algebraic expressions.

$\implies$ ${(2m+5n)}^2$ $\,=\,$ $2m \times (2m+5n)$ $+$ $5n \times (2m+5n)$

$\implies$ ${(2m+5n)}^2$ $\,=\,$ $2m \times 2m$ $+$ $2m \times 5n$ $+$ $5n \times 2m$ $+$ $5n \times 5n$

$\implies$ ${(2m+5n)}^2$ $\,=\,$ ${(2m)}^2$ $+$ $10mn$ $+$ $10nm$ $+$ ${(5n)}^2$

$\implies$ ${(2m+5n)}^2$ $\,=\,$ $4m^2$ $+$ $10mn$ $+$ $10mn$ $+$ $25n^2$

$\,\,\, \therefore \,\,\,\,\,\,$ ${(2m+5n)}^2$ $\,=\,$ $4m^2$ $+$ $20mn$ $+$ $25n^2$

Thus, the square of sum of two terms is expanded mathematically in terms of them to calculate its value. In this method, it takes six steps to find the value of square of the binomial. If you are a beginner, you must calculate the square of sum of two terms algebraically by the multiplications of algebraic expressions for practice.

Formula

An algebraic identity is derived in mathematics to find the square of sum of two terms directly. It is usually written in two ways popularly. You can follow any one of them.

$(1) \,\,\,$ ${(a+b)}^2$ $\,=\,$ $a^2+b^2+2ab$

$(2) \,\,\,$ ${(x+y)}^2$ $\,=\,$ $x^2+y^2+2xy$

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