$\mathrm{\omega }$ is a non real complex cube root of unity and $\mathrm{a},\mathrm{b}\in \mathrm{R}.$ If $\mathrm{\omega },{\mathrm{\omega }}^2$ are roots of $\frac{1}{\mathrm{a}+\mathrm{x}}+\frac{1}{\mathrm{b}+\mathrm{x}}=\frac{3}{\mathrm{x}}$ then a,b are roots of

$\begin{array}{rcl}\frac{1}{\mathrm{a}+\mathrm{x}}+\frac{1}{\mathrm{b}+\mathrm{x}}& =& \frac{3}{\mathrm{x}}\end{array}$

$\begin{array}{rcl}(\mathrm{b}+\mathrm{x}+\mathrm{a}+\mathrm{x}) \mathrm{x}& =& 3(\mathrm{a}+\mathrm{x}) (\mathrm{b}+\mathrm{x})\end{array}$

$\begin{array}{rcl}2{\mathrm{x}}^2+(\mathrm{a}+\mathrm{b})\mathrm{x}& =& 3(\mathrm{ab}+\mathrm{ax}+\mathrm{bx}+{\mathrm{x}}^2)\end{array}$

$\begin{array}{rcl}{\mathrm{x}}^2+2(\mathrm{a}+\mathrm{b})\mathrm{x}+3\mathrm{ab}& =& 0\end{array}$

$\begin{array}{rcl}\mathrm{w}+{\mathrm{w}}^2& =& -2(\mathrm{a}+\mathrm{b}), \mathrm{w}.{\mathrm{w}}^2=3\mathrm{ab}\end{array}$

$\begin{array}{rcl}-2(\mathrm{a}+\mathrm{b})& =& -1, {\mathrm{w}}^3=3\mathrm{ab}\end{array}$

$\begin{array}{rcl}\mathrm{a}+\mathrm{b}& =& \frac{1}{2}, \mathrm{ab}=\frac{1}{3}\end{array}$

a, b are roots of ${\mathrm{x}}^2-\mathrm{x}(\mathrm{a}+\mathrm{b})+\mathrm{ab}=0$

$\begin{array}{rcl} & \Rightarrow & {\mathrm{x}}^2-\mathrm{x}\left(\frac{1}{2}\right)+\frac{1}{3}=0\\ 6{\mathrm{x}}^2-3\mathrm{x}+2& =& 0\end{array}$