Question

$x^2+3ax+2a^2=0$
If the above equation has roots $\alpha ,\beta$ and it is given that ${\alpha }^2+{\beta }^2=5$, then the value of $a$ is

• A. $1$
• B. $-1$
• C. $\pm 1$
• D. None of these

Answer 1

The correct option is C $\pm 1$

Given equation is $x^2+3ax+2a^2=0$

We know $\alpha +\beta =-3a$ and $\alpha \beta =2a^2$
$\Rightarrow {\alpha }^2+{\beta }^2=5$ ...given

$\Rightarrow (\alpha +\beta {)}^2-2\alpha \beta ={\alpha }^2+{\beta }^2$
$\Rightarrow (\alpha +\beta {)}^2-2\alpha \beta =5$
$\Rightarrow 9a^2-2\left(2a^2\right)=5$
$\Rightarrow 5a^2=5$
$\Rightarrow a=\pm 1$