Factorize the following algebraic expression:

$9 z^{2} - x^{2} + 4 x y - 4 y^{2}$

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Solution

Given : $9 z^{2} - x^{2} + 4 x y - 4 y^{2}$

$= 9 z^{2} - (x^{2} - 4 x y + 4 y^{2})$

$= 9 z^{2} - [(x)^{2} - 2 (x) (2 y) + (2 y)^{2}]$

Using the formula, $(a - b)^{2} = a^{2} - 2 a b + b^{2}$

$= 9 z^{2} - (x - 2 y)^{2}$

$= (3 z)^{2} - (x - 2 y)^{2}$

Using, $a^{2} - b^{2} = (a + b) (a - b)$

$= [3 z - (x - 2 y)] [3 z + (x - 2 y)]$

$= (3 z - x + 2 y) (3 z + x - 2 y)$

$= (- x + 2 y + 3 z) (x - 2 y + 3 z)$

$∴ 9 z^{2} - x^{2} + 4 x y - 4 y^{2} = (- x + 2 y + 3 z) (x - 2 y + 3 z) .$

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9z² - x²+ 4xy - 4y²

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