Question

Let $\alpha, \beta$ be real roots of the quadratic equation $x ^2+ kx +\left( k ^2+2 k -4\right)=0$, then the maximum value of $\left(\alpha^2+\beta^2\right)$ is equal to

• A. 9
• B. 10
• C. 11
• D. 12

Answer 1

Answer: (D)

$\text { For real roots } D \geq 0 $
$k ^2-4\left( k ^2+2 k -4\right) \geq 0 \Rightarrow-3 k ^2-8 k +16 \geq 0 $
$\Rightarrow 3 k ^2+8 k -16 \leq 0 \Rightarrow 3 k ^2+12 k -4 k -16 \leq 0 $
$\Rightarrow 3 k ( k +4)-4( k +4) \leq 0 \Rightarrow k \in\left[-4, \frac{4}{3}\right] $
Also, $\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta=k^2-2\left(k^2+2 k-4\right)=-k^2-4 k+8=12-(k+2)^2$
Maximum value is 12 when $k =-2$.