If a, b, c are in GP where a, c are positive, then the equation ax² + bx + c = 0 has

The given equation is $a{x}^2+bx+c=0$

Given $a,b,c$ are in Geometric progression

It implies that $b^2=ac$ and given both $a,c$ are positive values

Use the quadratic formula 

           $\begin{array}{rcl}x& =& \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ & =& \frac{-b}{a}\left(\frac{-1\pm i\sqrt{3}}{2}\right)\\ & =& -\frac{b}{a}\omega ,\frac{-b}{a}{\omega }^2\end{array}$

$$\begin{gathered} {\mathrm{b}}^2=\mathrm{ac}\text{ and roots are }=\frac{\mathrm{b}}{\mathrm{a}}\left(\frac{-1\pm \sqrt{3\mathrm{i}}}{2}\right)=\frac{\mathrm{b}}{\mathrm{a}}\mathrm{\omega } \text{ and }\frac{\mathrm{b}}{\mathrm{a}}{\mathrm{\omega }}^2 \\ \mathrm{Ratio} \mathrm{of} \mathrm{the} \mathrm{roots} \mathrm{is} \mathrm{is} 1:\mathrm{\omega } \end{gathered}$$