If $a^{2} + b^{2} + c^{2} + 3 = 2 (a - b - c)$ , then the value of $2 a - b + c$ is:

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Solution

The correct option is D 2
$a^{2} + b^{2} + c^{2} + 3 = 2 (a - b - c)$
$a^{2} + b^{2} + c^{2} + 3 = 2 a - 2 b - 2 c$
$a^{2} + b^{2} + c^{2} + 3 - 2 a + 2 b + 2 c = 0$
$a^{2} - 2 a + b^{2} + 2 b + c^{2} + 2 c + 3 = 0$
$a^{2} - 2 a + 1 + b^{2} + 2 b + 1 + c^{2} + 2 c + 1 = 0$
$(a - 1)^{2} + (b + 1)^{2} + (c + 1)^{2} = 0$
$(a - 1)^{2} \geq 0$ $(b + 1)^{2} \geq 0$ $(c + 1)^{2} \geq 0$ $∵ x^{2} \geq 0$ $\forall x ϵ R$
$∴ (a - 1)^{2} = 0$ $(b + 1)^{2} = 0$ $(c + 1)^{2} = 0$
$a - 1 = 0$ $b + 1 = 0$ $c + 1 = 0$
$a = 1$ $b = - 1$ $c = - 1$
$∴ 2 a - b + c = 2 (1) - (- 1) + (- 1) = 2 + 1 - 1 = 2$

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