Let and lt be the roots of the equation x2 bx c 0 where b gt...........

Mathematics (NDA) Quadratic Equations

Let $\small \alpha$ and $\small \beta$ ( $\small \alpha$ < $\small \beta$ ) be the roots of the equation x²+ bx+ c = 0, where b > 0 and c < 0.

Consider the following

1. $\small \alpha$ + $\small \beta$ + $\small \alpha$ $\small \beta$ > 0

2. $\small \alpha$ ² $\small \beta$ + $\small \beta$ ² $\small \alpha$ > 0

Which of the above is/are correct?

Only 1

Only 2

Both 1 and 2

Neither 1 nor 2

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Explanation:

Given, $\small \alpha$ and $\small \beta$   are the roots of equation x² + bx + c = 0.
$\small \therefore$    $\small \alpha$ + $\small \beta$ = − b and $\small \alpha$ $\small \beta$ = c
As b > 0
So, $\small \alpha$ + $\small \beta$ < 0
=>   $\small \beta$ < − $\small \alpha$
and given that $\small \alpha$ < $\small \beta$   => $\small \alpha$ < 0 and $\small \beta$ > 0