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

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Solution

Factorise the given expression $25 x^{2} + 4 y^{2} + 9 z^{2} - 20 x y - 12 y z + 30 x z$
To factorize this we use identity $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a .$
The given expression using identity can be rewritten as:
$25 x^{2} + 4 y^{2} + 9 z^{2} - 20 x y - 12 y z + 30 x z = {(5 x)}^{2} + {(- 2 y)}^{2} + {(3 z)}^{2} + 2 (5 x) (- 2 y) + 2 (- 2 y) (3 z) + 2 (3 z) (5 x) = {(5 x + (- 2 y) + 3 z)}^{2}$
Hence $25 x^{2} + 4 y^{2} + 9 z^{2} - 20 x y - 12 y z + 30 x z = {(5 x - 2 y + 3 z)}^{2}$ .

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