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

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Solution

We have: $9 x^{2} + 16 y^{2} + 4 z^{2} - 24 x y + 16 y z - 12 x z$
$= (- 3 x)^{2} + (4 y)^{2} + (2 z)^{2} + 2 (- 3 x) (4 y) + 2 (4 y) (2 z) + 2 (2 z) (- 3 x)$
So, it is the form of $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
By replacing $a = - 3 x, b = 4 y, c = 2 z$ in the above formula, we get
$(- 3 x)^{2} + (4 y)^{2} + (2 z)^{2} + 2 (- 3 x) (4 y) + 2 (4 y) (2 z) + 2 (2 z) (- 3 x) = (- 3 x + 4 y + 2 z)^{2}$
Therefore, the factors of the given expression is $= (- 3 x + 4 y + 2 z)^{2}$

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