If $α$ and $β$ are the roots of $a x^{2} + b x + c = 0$ , Find the values of following.
$\frac{β}{a α + b} + \frac{α}{a β + b}$

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Solution

Since $α$ and $β$ are the roots of
$a x^{2} + b x + c = 0$
$a α^{2} + b α + c = 0 o r a α + b = \frac{c}{α} .$
$a β^{2} + b β + c = 0 o r a β + b = - \frac{c}{β} .$
Also $α + β = - \frac{b}{a}, α β . = \frac{c}{a} .$
$\frac{β}{a α + b} + \frac{α}{a β + b} = \frac{α β}{c} - \frac{α β}{c} = - \frac{2}{c}, \frac{c}{a} = - \frac{2}{a} .$

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