If $c \neq 0$ and $α, β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then value of $\frac{a α^{2} + c}{a α + b} + \frac{a β^{2} + c}{a β + b}$ is

\frac{b (b^{2} - 2 a c)}{4 a}

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

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

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

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Solution

The correct option is C $\frac{b (b^{2} - 2 a c)}{a^{2} c}$
$a α^{2} + b α + c = 0$
$a α^{2} + c = - b α$
$a α^{2} + b α = - c$
$a α + b = - \frac{c}{α}$
$∴ \frac{a α^{2} + c}{a α + b} = \frac{b}{c} α^{2}$
Similarly, $\frac{a β^{2} + c}{a β + b} = \frac{b}{c} β^{2}$ $\frac{a α^{2} + c}{a α + b} + \frac{a β^{2} + c}{a β + b} = \frac{b}{c} (α^{2} + β^{2}) = \frac{b}{c} (\frac{b^{2}}{a^{2}} - \frac{2 c}{a})$
$= \frac{(b^{2} - 2 a c) b}{a^{2} c}$

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