Factorize: $4 x^{2} + 9 y^{2} + 25 z^{2} - 12 x y - 30 y z + 20 z x$

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Solution

We have
$4 x^{2} + 9 y^{2} + 25 z^{2} - 12 x y - 30 y z + 20 z x$
$= [(2 x)^{2} + (- 3 y)^{2} + (5 z)^{2} + 2 (2 x) (- 3 y) + 2 (- 3 y) (5 z) + 2 (5 z) (2 x)]$
Comparing with the identity V,
$(x + y + z)^{2} \equiv x^{2} + y^{2} + z^{2} + 2 x y + 2 y z + 2 z x$ , we get
$= (2 x - 3 y + 5 z)^{2}$
$= (2 x - 3 y + 5 z) (2 x - 3 y + 5 z)$ .

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