Question

The condition that one root is twice the other root of the quadratic equation $x^2+px+q=0$ is

• A. $2p^2=9q$
• B. $3p^2=8q$
• C. $p^2=9q$
• D. $4p^2=q$

Answer 1

The correct option is A $2p^2=9q$

Let $\alpha$ and $2\alpha$ be the roots of given equation $x^2+px+q=0$

sum of roots $=-p$ and product of roots $=q$

$\Rightarrow \alpha +2\alpha =-p$ and $\alpha \times 2\alpha =q$

$\Rightarrow 3\alpha =-p$ $\Rightarrow \alpha =\frac{-p}{3}$ ....(i)

and

$2{\alpha }^2=q$ .....(ii)

put $\alpha =\frac{-p}{3}$ in equation (ii)

$\Rightarrow 2\times (\frac{-p}{3}{)}^2=q$

$\Rightarrow 2p^2=9q$

Option A is correct.