If $a, b, c$ are in GP, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in...
From the question, we know $b^2=ac \implies b=\sqrt{ac}$ $$ax^2+2\sqrt{ac}x+c= (\sqrt{a}x + \sqrt{c})^2=0$$ $$\therefore x=-\sqrt{\frac{c}{a}}$$
But from here, I do not know how to proceed. Please help.