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Factorize the expression $16(2x-1)^2-25y^2$.
Given:
The given expression is $16(2x-1)^2-25y^2$.
To do:
We have to factorize the expression $16(2x-1)^2-25y^2$.
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.
$16(2x-1)^2-25y^2$ can be written as,
$16(2x-1)^2-25y^2=[4(2x-1)]^2-(5y)^2$ [Since $16=4^2, 25=5^2$]
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,
$16(2x-1)^2-25y^2=[4(2x-1)]^2-(5y)^2$
$16(2x-1)^2-25y^2=[4(2x-1)+5y][4(2x-1)-5y]$
$16(2x-1)^2-25y^2=[4(2x)-4(1)+5y][4(2x)-4(1)-5y]$
$16(2x-1)^2-25y^2=(8x-4+5y)(8x-4-5y)$
$16(2x-1)^2-25y^2=(8x+5y-4)(8x-5y-4)$
Hence, the given expression can be factorized as $(8x+5y-4)(8x-5y-4)$.
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