If $a, b, c$ are real numbers in G.P. such that $a$ and $c$ are positive, then the roots of the equation $a{x}^2+bx+c=0$

We have, $b^2=ac$

Let, $a, \beta$ be the roots of the equation $a{x}^2+bx+c=0.$ Then,

$\begin{array}{l}\alpha , \beta =\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\alpha , \beta =\frac{-\sqrt{ac}\pm i\sqrt{3ac}}{2a} \left[\because b^2=ac\right] \end{array}$

$=\alpha =\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)\sqrt{\frac{c}{a}}$ and, $\beta =\left(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)\sqrt{\frac{c}{a}}$

$=\alpha =\omega \sqrt{\frac{c}{a}}$ and, $\beta ={\omega }^2\sqrt{\frac{c}{a}}\Rightarrow \alpha :\beta =1:\omega$