Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha$ and $4\beta$ be roots of $x^2-6x+a=0$ and $\alpha ,3\beta$ be the roots of $x^2-cx+6=0$, then $\alpha +4\beta =6$ and $4\alpha \beta =a$ $\alpha +3\beta =c$ and $3\alpha \beta =6$. We get $\alpha \beta =2\Rightarrow a=8$ So the first equation is $x^2-6x+8=0\Rightarrow x=2,4$ If $\alpha =2$ and $4\beta =4$ then $3\beta =3$ If $\alpha =4$ and $4\beta =2$, then $3\beta =3/2$ (non-integer) $\therefore$ common root is $x=2$.