Factorize the expression $27x^2-12y^2$.

Given:

The given expression is $27x^2-12y^2$.

To do:

We have to factorize the expression $27x^2-12y^2$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression means 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.

$27x^2-12y^2$ can be written as,

$27x^2-12y^2=3[9x^2-4y^2]$                (Taking $3$ as common)

$27x^2-12y^2=3[(3x)^2-(2y)^2]$             [Since $9=3^2, 4=2^2$]

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,

$27x^2-12y^2=3[(3x)^2-(2y)^2]$

$27x^2-12y^2=3(3x+2y)(3x-2y)$

Hence, the given expression can be factorized as $3(3x+2y)(3x-2y)$.