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Quantitative Aptitude

Quadratic Equations

If the roots ...

Question

If the roots of the equation $x^{2} + 2 b x + c = 0$ are $α$ and $β$ , then $b^{2} - c$ is equal to

A

\frac{(α - β)^{2}}{4}

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B

(α + β)^{2} - α β

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C

(α + β)^{2} + α β

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D

\frac{(α - β)^{2}}{2} + α β

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E

\frac{(α + β)^{2}}{4} - α β

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Solution

The correct option is E $\frac{(α + β)^{2}}{4} - α β$
Given equation is $x^{2} + 2 b x + c = 0$

Also $α$ and $β$ are the roots of the given equation.

Thus $α + β = \frac{- 2 b}{1} = - 2 b$

$\Rightarrow b = - \frac{α + β}{2}$ .......(i)

and $α \times β = \frac{c}{1} = c$ .......(ii)

Thus $b^{2} - c = {[- (\frac{α + β}{2})]}^{2} - α \times β$

$= \frac{(α + β)^{2}}{4} - α β$

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