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Definition of Functions

If α, β are...

Question

If $α, β$ are the roots of the equation $a x^{2} + b x + c = 0$ then show that $\frac{1}{a α + b} + \frac{1}{a β + b} = \frac{b}{a c}$

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Solution

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

then $a α^{2} + b α + c = 0$ and $a β^{2} + b β + c = 0$

$α (a α + b) + c = 0$ and $β (a β + b) + c = 0$

$\Rightarrow (a α + b) = \frac{- c}{α}$ and $(a β + b) = \frac{- c}{β}$

L.H.S $= \frac{1}{a α + b} + \frac{1}{a β + b}$

$= \frac{α}{- c} + \frac{β}{- c}$

$= \frac{- α - β}{c}$

$= \frac{- (α + β)}{c}$

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

$= \frac{b}{a c}$

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