If and are the roots of the quadratic equation 2x2 6x k 0 where k 0 then what is...........

Quantitative Aptitude (CDS) Quadratic Equations

If $\alpha$ and $\beta$ are the roots of the quadratic equation 2x² + 6x + k = 0, where k 0,then what is the maximum value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }\;?$

-2

-9

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Explanation:

As $\alpha$ and $\beta$ are the roots of equation $\alpha$ and $\beta$
2x² + 6x + k = 0, ?? < 0 ………. (i)
Now,
$\alpha$ / $\beta$ + $\beta$ / $\alpha$ =( $\alpha$ ²+ $\beta$ ²)/ $\alpha$ $\beta$ ………. (ii)
From (i)
Sum of roots = −6/2= −3
Products of roots =k/2
Now, ( $\alpha$ + $\beta$ )² = $\alpha$ ² + $\beta$ ² + 2 $\alpha$ $\beta$
(−3)² = $\alpha$ ² + $\beta$ ² +2k/2
$\alpha$ ² + $\beta$ ² =9-k From (ii)