Question

If the two equations $x^2-11x+a=0$ and $x^2-14x+2a=0$ have a common root and $a\ne 0,$ then the value of the common root is

Answer 1

Let $\alpha$ be the common root.
${\alpha }^2-11\alpha +a=0...\left(1\right){\alpha }^2-14\alpha +2a=0...\left(2\right)$
Subtracting $\left(1\right)$ and $\left(2\right)$, we get
$\alpha =\frac{a}{3}$
Substituting $\alpha =\frac{a}{3}$ in equation $\left(1\right)$,
$\frac{a^2}{9}-\frac{11a}{3}+a=0\Rightarrow a^2-24a=0\Rightarrow a=24(\because a\ne 0)\therefore \alpha =8$