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

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\frac{- 2}{a}

\frac{- 2}{b}

\frac{- 1}{a}

\frac{- 1}{b}

answer is A.

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Detailed Solution

As per the question, α and β are the zeros or roots of the quadratic polynomial f(x)=

{ax}^{2} + bx + c

, this is a general quadratic equation so, their roots are also general.
Therefore, as per the given equation sum and product of these roots will be as below:
As we know the sum of the root of an equation, α+β=

- \frac{b}{a}

And product of roots, αβ=

\frac{c}{a}

Now, we will simplify the equation which we want to evaluate by taking L.C.M.

\frac{β}{\propto \propto + b} + \frac{\propto}{\propto β + b}

While taking L.C.M. In numerator, we will multiply β with (aβ+b), α with (aα+b) and in denominator we will multiply (aα+b)(aα+b) with (aβ+b)
Here, the result of L.C.M.
=

\frac{a β^{2} + βb + a α^{2} + bα}{a^{2} αβ + aαb + abβ + b^{2}}

Now, we will take commons from the numerator: a common from first, third terms and take b common from second, fourth terms.
Take commons from the denominator: ab second and third terms.
After that we get =