Use the relation $x^3=y$ to get $\alpha^3 + \beta^3+\gamma^3$

Hint: $$y+0.5=y^{\frac{2}{3}}+y^{\frac{1}{3}}$$ Now cube both sides to get a cubic equation in $y$. $$\Longrightarrow y^3+1.5 y^2+ 0.75 y+0.125 = y^2+y+3y(y^{\frac{2}{3}}+y^{\frac{1}{3}})$$

Now use the first equation again in it.