Factorize: 4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^2 - Vidyadip

Question

Factorize:
$4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^2$

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Answer

$4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^{2}$
Let $(x - y) = x,(x + y) = y = 4x^2 - 12xy + 9y^2$
Splitting the middle term $-12 = -6 - 6$
also$ 4 \times 9 = -6 \times -6 = 4x^2 - 6xy - 6xy + 9y^2$
$= 2x(2x - 3y) - 3y(2x - 3y)$
$= (2x - 3y)(2x - 3y) = (2x - 3y)^2$
Substituting $x = x - y$ & $y$
$= x + y = [2(x - y) - 3(x + y)]^2 = [2x - 2y - 3x - 3y]^2$
$= (2x - 3x - 2y - 3y)² = [-x - 5y]^2$
$= [(-1)(x + 5y)]^2 = (x + 5y)^2 [(-1)^2 = 1]$
$\therefore$ $4(x - y)^2 - 12(x - y)(x + y) + 9(x + y)^2 = (x + 5y)^2$

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