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

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Solution

We have,
$9 x^{2} + 4 y^{2} + 16 z^{2} - 12 x y - 16 y z + 24 x z$
$= (3 x)^{2} + (- 2 y)^{2} + (4 z)^{2} + 2 (3 x) (- 2 y) + 2 (- 2 y) (4 z) + 2 (3 x) (4 z)$
$[∵ a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a = (a + b + c)^{2}]$
$= (3 x - 2 y + 4 z)^{2}$
$= (3 x - 2 y + 4 z) (3 x - 2 y + 4 z)$

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