Factorize $4 y^{2} - 12 y z - x^{2} + 9 z^{2}$

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Solution

$4 y^{2} - 12 y z - x^{2} + 9 z^{2}$

$= 4 y^{2} - 12 y z + 9 z^{2} - x^{2}$

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

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

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

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

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$16 x^{2} + 4 y^{2} + 9 z^{2} - 16 x y - 12 y z + 24 x z$

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

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