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

The expansion of a plus b whole squared identity can be derived in algebraic mathematics by multiplying the binomial $a+b$ with the same binomial, using the multiplication of algebraic expressions.

### Product of Special Binomials

Multiply the binomial $a+b$ by itself to express the product of them in mathematical form.

$(a+b) \times (a+b)$

According to the exponentiation, the product of two same binomials can be written in exponential notation.

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

It is a special case in algebra. Hence, the product of two same sum basis binomials is often called as the special product of binomials or the special binomial product.

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

### Multiplication of Special Binomials

The $a$ plus $b$ whole square represents the square of sum of two terms and it can be expanded by multiplying the algebraic expression $a+b$ with same binomial. Therefore, multiply the algebraic expressions by using multiplication of algebraic expressions.

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

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

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

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

### Simplify the Algebraic expression

The square of sum of terms $a$ and $b$ is expanded as an algebraic expression $a^2+ab+ba+b^2$. According to commutative property, the product of $a$ and $b$ is equal to the product of $b$ and $a$. So, $ab = ba$.

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

Now, add the like algebraic terms in the algebraic expression by the addition of algebraic terms for simplifying the expansion of the $a$ plus $b$ whole square identity.

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

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

Algebraically, it is proved that the $a$ plus $b$ whole square is equal to the $a$ squared plus $b$ squared plus two times product of $a$ and $b$. It is used as formula in mathematics to expand square of sum of two terms.

Latest Math Topics

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

###### Maths Topics

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

###### Maths Problems

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

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.