If the expression $x^{2} + 2 (a + b + c) x + 3 (b c + c a + a b)$ is a perfect square, then

a = b = c

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a = \pm b = \pm c

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a = b \neq c

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None of the above

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Solution

The correct option is A $a = b = c$
$x^{2} + 2 (a + b + c) x + 3 (a b + b c + a c) i s a p e r f e c t s q u a r e ∴ D = 0 4 {(a + b + c)}^{2} - 4.3 (a b + b c + a c) = 0 a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c) - 3 (a b + b c + a c) = 0 a^{2} + b^{2} + c^{2} - a b - b c - a c = 0 ∴ \frac{1}{2} [{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}] = 0 ∴ a = b = c A s s u m o f s q u a r e s i s z e r o o n l y i f s q u a r e a r e z e r o$

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