Factorize: a^2 + 4b^2 - 4ab - 4c^2 - Vidyadip

Question

Factorize:
$a^2 + 4b^2 - 4ab - 4c^2$

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Answer

The given expression to be factorized is:
$a^2 + 4b^2 - 4ab - 4c^2$
This can be arrange in the form
$a^2 + 4b^2 - 4ab - 4c^2$
$= (a^2 - 4ab + 4b^2) - 4c^2$
$= {(a)^2 - 2.a.2b + (2b)^2} - 4c^2$
$= (a - 2b)^2 - 4c^2$
Substitute $x = (a - 2b)$
$a^2 + 4b^2 - 4ab - 4c^2 = x^2 - 4c^2$
$= x^2 - (2c)^2$
$= (x + 2c)(x - 2c)$
$Put x = (a - 2b)$
$a^2 + 4b^2 - 4ab - 4c^2 = {(a - 2b) + 2c}{(a - 2b) - 2c}$
$= (a - 2b + 2c)(a - 2b - 2c)$
we cannot further factorize the expresion.
So, the required factorization of $a^2 + 4b^2 - 4ab - 4c^2 is (a - 2b + 2c)(a - 2b - 2c)$

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