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
$x^2-6x+c=0$
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 $c=(2{)}^3=8$