Simplify : $(a + b + c)^{2} + (a - b + c)^{2} + (a + b - c)^{2}$

3 (a^{2} + b^{2} + c^{2}) + 2 (b c - a b - a c)

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3 (a^{2} + b^{2} + c^{2}) + 3 (b c - a b - a c)

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3 (a^{2} + b^{2} + c^{2}) - 2 (b c + a b + a c)

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3 (a^{2} + b^{2} + c^{2}) - 2 (b c - a b - a c)

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Solution

The correct option is D $3 (a^{2} + b^{2} + c^{2}) - 2 (b c - a b - a c)$
The given expression $(a + b + c)^{2} + (a - b + c)^{2} + (a + b - c)^{2}$ can be simplified as follows:

$(a + b + c)^{2} + (a - b + c)^{2} + (a + b - c)^{2} = (a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a) + [a^{2} + (- b)^{2} + c^{2} + 2 a (- b) + 2 (- b) c + 2 c a]$
$+ [a^{2} + b^{2} + (- c)^{2} + 2 a b + 2 b (- c) + 2 (- c) a] (∵ (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a) = (a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a) + (a^{2} + b^{2} + c^{2} - 2 a b - 2 b c + 2 c a)$
$+ (a^{2} + b^{2} + c^{2} + 2 a b - 2 b c - 2 c a) = 3 a^{2} + 3 b^{2} + 3 c^{2} + 2 a b - 2 b c + 2 c a = 3 (a^{2} + b^{2} + c^{2}) - 2 (b c - a b - a c)$

Hence, $(a + b + c)^{2} + (a - b + c)^{2} + (a + b - c)^{2} = 3 (a^{2} + b^{2} + c^{2}) - 2 (b c - a b - a c)$

Suggest Corrections

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Factorise $b^{2} + c^{2} + 2 b c - a^{2}$

\frac{a^{2} - {(b - c)}^{2}}{{(a + c)}^{2} - b^{2}} + \frac{b^{2} - {(a - c)}^{2}}{{(a + b)}^{2} - c^{2}} + \frac{c^{2} - {(a - b)}^{2}}{{(b + c)}^{2} - a^{2}}

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