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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 $3(3+7a)(1-a)$.
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