Factorize the expression $(3+2a)^2-25a^2$.

Given:

The given algebraic expression is $(3+2a)^2-25a^2$.

To do:

We have to factorize the expression $(3+2a)^2-25a^2$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution. 

An algebraic expression is factored completely when it is written as a product of prime factors.

$(3+2a)^2-25a^2$ can be written as,

$(3+2a)^2-25a^2=(3+2a)^2-(5a)^2$             [Since $25=(5)^2$]

Here, we can observe that the given expression is a difference of two squares. So, by using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize the given expression. 

Therefore,

$(3+2a)^2-25a^2=(3+2a)^2-(5a)^2$

$(3+2a)^2-25a^2=(3+2a+5a)(3+2a-5a)$

$(3+2a)^2-25a^2=(3+7a)(3-3a)$

$(3+2a)^2-25a^2=(3+7a)3(1-a)$                    (Taking $3$ common)

$(3+2a)^2-25a^2=3(3+7a)(1-a)$

Hence, the given expression can be factorized as $3(3+7a)(1-a)$.

Akhileshwar Nani

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Updated on: 08-Apr-2023

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