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Factorize the expression $9(a-b)^2-100(x-y)^2$.
Given:
The given expression is $9(a-b)^2-100(x-y)^2$.
To do:
We have to factorize the expression $9(a-b)^2-100(x-y)^2$.
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.
$9(a-b)^2-100(x-y)^2$ can be written as,
$9(a-b)^2-100(x-y)^2=[3(a-b)]^2-[10(x-y)]^2$ [Since $9=3^2, 100=(10)^2$]
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,
$9(a-b)^2-100(x-y)^2=[3(a-b)]^2-[10(x-y)]^2$
$9(a-b)^2-100(x-y)^2=[3(a-b)+10(x-y)][3(a-b)-10(x-y)]$
$9(a-b)^2-100(x-y)^2=(3a-3b+10x-10y)(3a-3b-10x+10y)$
Hence, the given expression can be factorized as $(3a-3b+10x-10y)(3a-3b-10x+10y)$.
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