If $\text{⨍}\left(\mathrm{x}\right) = {\mathrm{x}}^3 +{\mathrm{\alpha x}}^2 +\mathrm{\beta x}+\mathrm{\gamma }$ where $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are rational numbers and two roots $\text{⨍}\left(\mathrm{x}\right) = 0$ are eccentricities of a parabola and a rectangular hyperbola then $\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }$ is equal to =

$$\begin{gathered} \mathrm{eccentricity} \mathrm{of} \mathrm{parabola}=1, \\ \mathrm{eccenticity} \mathrm{of} \mathrm{rectengular} \mathrm{hyperbola}=\sqrt{2} \\ \mathrm{then} \mathrm{third} \mathrm{root} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is}= -\sqrt{2} \\ \therefore \mathrm{f}\left(\mathrm{x}\right)=(\mathrm{x}-1) (\mathrm{x}-\sqrt{2}) (\mathrm{x}+\sqrt{2}) \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3-{\mathrm{x}}^2-2\mathrm{x}+2 \\ \mathrm{By} \mathrm{compari}ng \mathrm{with} \mathrm{given} \mathrm{polynomia}l we get \\ \mathrm{\alpha }=-1, \mathrm{\beta }=-2, \mathrm{\gamma }=2 \\ \mathrm{Now} \mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=-1 \end{gathered}$$