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

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Solution

If $α$ and $β$ are roots / zero of the quadratic equation
$f (x) = a x^{2} + b x^{2} + c$

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

observe, we have
$α + β = \frac{- b}{a}$
$α β = \frac{c}{a}$
$\Rightarrow α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= \frac{b^{2}}{a^{2}} - \frac{2 c}{a}$

Using equations on the right

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

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

$= \frac{- 2 c}{a c} = \frac{- 2}{a}$

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if α and βare zeroes of polynomial ax2+bx+c then find 1/aα+b + 1/aα+b

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

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