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Definition of Functions

If α, β are...

Question

If $α, β$ are the zeroes of the polynomial $f (x) = a x^{2} + b x + c$ , then find $\frac{1}{α} + \frac{1}{β} - 2 α β$

\frac{b + 2 a c}{2 a}

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- (\frac{b}{a} + \frac{c}{a b})

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

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(\frac{b}{c} + \frac{a}{b})

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Solution

The correct option is A $- (\frac{b}{c} + \frac{2 c}{a})$
Given, $α, β$ are the zeroes of the polynomial $f (x) = a x^{2} + b x + c$
$\Rightarrow α + β = - \frac{b}{a}$
$\Rightarrow α β = \frac{c}{a}$
Now, $\frac{1}{α} + \frac{1}{β} - 2 α β = \frac{β + α}{β α} - 2 α β$
$= \frac{- \frac{b}{a}}{\frac{c}{a}} - 2 \frac{c}{a}$
$= \frac{- b}{c} - 2 \frac{c}{a}$
$= - (\frac{b}{c} + \frac{2 c}{a})$
Option C is correct.

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