Determining the value of required parameter for the equations to have a common root 
Determine the value of $a$ such that $x^2-11x+a=0$ and $x^2-14x+2a=0$ may have a common root.

My attempt: 
Let the common root be $\alpha$
On substituting $\alpha$ in both equations and then subtracting, $a = -3\alpha$
How do I continue from here? What are the other conditions for them to have common roots? 

If you do the calculation correctly $a=3\alpha$. So,$x^2-11*x+3\alpha=0$ is the first equation and it has root $\alpha$. So $\alpha^2-11*\alpha+3\alpha=0$, that is $\alpha(\alpha-8)=0$. So, $\alpha=0,8$. If common root $\alpha=8$ then $a=3\alpha=3*8=24$. If $\alpha=0$ then $a=0$.
