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If the equation $a x^{2} + b x + c = 0, c x^{2} + b x + a = 0, a \neq c$ have a negative common root, then the value of $a - b + c$ is

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Detailed Solution

a x^{2} + b x + c = 0 & c x^{2} + b x + c = 0

have a negative common root then

${(c^{2} - a^{2})}^{2} = (a b - b c) (a b - b c)$

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

${(c + a)}^{2} - b^{2} = 0$

$(c + a + b) (a + c - b) = 0$

$∴ a - b + c = 0$

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