Question

For what values of $a$ do the equations $x^2+ax+1=0$ and $x^2+x+a$ have a root in common?

Answer 1

$x^2+ax+1=0$

$x^2+x+a=0$

Let common root be x.

${\alpha }^2+\alpha a+1=0$-------(1)

${\alpha }^2+\alpha +a=0$---------(2)

(1)-(2)

$\alpha (a-1)+(1-a)=0$

$\alpha =1$

Putting $\alpha =1$ in equation $1$

$1^2+1a+1=0$

So, $a=-2$