Math Doubts

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.

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
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