Math Doubts

Proof of ${(a+b)}^2$ formula in Algebraic Method


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


$a$ and $b$ are two literal numbers and the summation of them is $a+b$. It is known as a binomial and the square of this binomial is expressed as ${(a+b)}^2$. The rule for expanding this is called $a+b$ whole square formula in algebra.


Multiplying Binomials

The square of the binomial $a+b$ can be expanded algebraically by the multiplications of the two same binomials.

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

Apply the multiplication of algebraic expressions rule.

$\implies {(a+b)}^2$ $\,=\,$ $a \times (a+b) +b \times (a+b)$

$\implies {(a+b)}^2$ $\,=\,$ $a \times a + a \times b + b \times a + b \times b$

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


Identifying the Like terms

Mathematically, The product $a$ and $b$ is equal to the product of $b$ and $a$. So, $ab = ba$.

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

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


Adding Like terms

There are two $ab$ terms in the expansion of the square of the sum of the terms. So, they can be added algebraically on the basis of addition of algebraic terms.

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

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

In this way, the $a+b$ whole square identity is proved in algebraic approach.

Therefore, it has been proved that $a+b$ whole square is equal to $a$ squared plus $b$ squared plus plus two times the product of $a$ and $b$.

Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

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.

Learn more