Expand: ${(3 a - 2 b + 5 c)}^{2}$

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Solution

We have ${(x + y + z)}^{2} = x^{2} + y^{2} + z^{2} + 2 (x y + y z + z x)$

Take $x = 3 a, y = - 2 b$ and $z = 5 c$

${(3 a - 2 b + 5 c)}^{2} = {(3 a)}^{2} + {(- 2 b)}^{2} + {(5 c)}^{2} + 2 (3 a \times - 2 b + (- 2 b) \times 5 c + 5 c \times 3 a)$

$= 9 a^{2} + 4 b^{2} + 25 c^{2} + 2 (- 6 a b - 10 b c + 15 a c)$

$= 9 a^{2} + 4 b^{2} + 25 c^{2} - 12 a b - 20 b c + 30 a c$

$∴ {(3 a - 2 b + 5 c)}^{2} = 9 a^{2} + 4 b^{2} + 25 c^{2} - 12 a b - 20 b c + 30 a c$

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