Question

Find the value of $\lambda$, so that the equations $x^2-x-12=0$ and $\lambda x^2+10x+3=0$ may have one root in common.Also, find the common root.

Answer 1

$\because x^2-x-12=0$

$\Rightarrow (x-4)(x+3)=0$

$\therefore x=4,-3$

If $x=4$ is a common root, then

$\lambda {\left(4\right)}^2+10\times 4+3=0$

$\Rightarrow 16\lambda =-43$

$\therefore \lambda =\frac{-43}{16}$

and if $x=-3$ is a common root, then

$\lambda {(-3)}^2+10\times -3+3=0$

$\Rightarrow 9\lambda =27$

$\therefore \lambda =\frac{27}{9}=3$

Hence for $\lambda =\frac{-43}{16}$, common root is $x=4$

and for $\lambda =3,$ common root is $x=-3$