If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = a x^{2} + b x + c$ , then evaluate :
$\frac{1}{a α + β} + \frac{1}{a β + α}$

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Solution

Given,

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

$α + β = - \frac{b}{a}$

$α β = \frac{c}{a}$

$\frac{1}{a α + b} + \frac{1}{a β + b}$

$= \frac{a β + b + a α + b}{(a α + b) (a β + b)}$

$= \frac{a (α + β) + 2 b}{a^{2} α β + a b (α + β) + b^{2}}$

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

$= \frac{- b + 2 b}{a c - b^{2} + b^{2}} = \frac{b}{a c}$

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