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

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Solution

If $α$ and $β$ was of $f (x) = a x^{2} + b x + c$ . Then evaluate
$\frac{1}{a α + b} + \frac{1}{a β + b}$
$\frac{(a β + b) + (a α + b)}{(a α + b) (a β + b)}$
$= \frac{a (α + β) + 2 b}{a^{2} α β + a b α + a b β + b^{2}}$
$= \frac{a (α + β) + 2 b}{a^{2} (α β) + a b (α + β) + b^{2}}$
Using equation on the right side we get
$= \frac{a (- b / a) + 2 b}{a^{2} (c / a) + a b (- b / a) + b^{2}} = \frac{- b + 2 b}{a c - b^{2} + b^{2}} = \frac{b}{a c}$
observe that, we have
$α + β = \frac{- b}{a}$
$α β = \frac{c}{a}$

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