Question

If $\alpha$ & $\beta$ are the zeroes of quad. polynomial $x^2-(k+6)x+2(2k-1)$. Find the value of $k$ so that $\alpha +\beta =\frac{1}{2}\alpha \beta$

Answer 1

$\alpha ,\beta$ are the zeroes of $x^2-(k+6)x+2(2k-1)$

i.e, $\alpha +\beta =k+6$ ( Sum of roots of quadratic )

and $\alpha \beta =2(2k-1)$ ( Product of roots )

According to question, $\alpha +\beta =\frac{1}{2}\alpha \beta$

$\Rightarrow k+6=\frac{2(2k-1)}{2}=2k-1$

$\Rightarrow k+6=2k-1$

$\Rightarrow k=7$

Hence, the answer is $7.$