Question

If $\alpha$ and $\alpha^2$ are the roots of the equation $x^2 + 6x + c = 0$, then the positive value of $c$ is

• A. 2
• B. 3
• C. 4
• D. 9
• E. 8

Answer 1

Answer: (E)

Given equation is
$x^{2}-6 x+c=0$
Roots of the equation are $\alpha$ and $\alpha^{2}$.
$\therefore \alpha+\alpha^{2}=6 \,...(i)$
and $\alpha \cdot \alpha^{2}=c$
$\Rightarrow \alpha^{3}=c \,....(ii)$
From Eq. (i), we have
$\alpha+\alpha^{2}-6=0$
$\Rightarrow \alpha^{2}+\alpha-6=0$
$\Rightarrow (\alpha+3)(\alpha-2)=0$
$\Rightarrow \alpha=-3,2$
$\because$ Value of $c$ is positive.
$\therefore \alpha=2$ and $c=(2)^{3}=8$