Question

Let $x^2-px+q=0$, where $p\in R,q\in R$ have the roots $\alpha ,\beta$ such that $\alpha +2\beta =0$, then

• A. $2p^2+q=0$
• B. $2q^2+p=0$
• C. $q<0$
• D. $p=q$

Answer 1

Answer: (A), (C)

The correct options are
A $2p^2+q=0$
C $q<0$

$\alpha +\beta =p$
$\alpha \beta =q$
$\alpha +2\beta =0$
$\Rightarrow p+\beta =0$
$\Rightarrow \beta =-pand\alpha =2p$
$\Rightarrow q=-2p^2<0$
Substituting $\beta =-pin{x}^2-px+q=0$
$\Rightarrow 2p^2+q=0$