If the equations x² + ax + bc = 0 and x² + bx + ca = 0 have a common root, then their other roots satisfy the equation

Let $\alpha$ be the common root 

$$\begin{gathered} {\alpha }^2+a\alpha +bc=o.....\left(1\right) \\ {\alpha }^2+b\alpha +ca=o....\left(2\right) \\ \therefore \left(1\right)-\left(2\right)\Rightarrow \left(a-b\right)\alpha +c\left(b-a\right)=o\Rightarrow \alpha =c \\ a,b are the other roots \\ The required equation is x^2-\left(a+b\right)x+ab=0,and a+b+c=0 \\ \Rightarrow x^2+cx+ab=0 \end{gathered}$$