Question

The positive value of $\lambda$ for which the equations $x^2-x-12=0$ and $\lambda x^2+10x+3=0$ have one root in common, is

Answer 1

$x^2-x-12=0$ and $\lambda x^2+10x+3=0$ have a common root.
$x^2-x-12=0\Rightarrow (x+3)(x-4)=0\Rightarrow x=4,-3$

Case $1:$
$x=4$ is a common root.
$\Rightarrow 16\lambda +43=0$
$\Rightarrow \lambda =-\frac{43}{16}$

Case $2:$
$x=-3$ is a common root.
$\Rightarrow 9\lambda -27=0$
$\Rightarrow \lambda =3$