solving a system of cubic equations with real roots Consider the following system of equations:
\begin{align}
\frac{1}{8} (\alpha +2 x)^2 \left((\alpha +2 x)^2-12\right)+\\ +\frac{4}{9} y^2 \left((\alpha
   +2 x)^2+2\right)+\frac{32 y^4}{81}&=0\\
\alpha ^3-6 \alpha +8 x^3-4 \alpha  x^2-2 \left(\alpha ^2+2\right) x&=0
\end{align}
I would like to eliminate the $\alpha$ value in order to have an equation of the form $f(x,y)=0$.
Using Cardano's formula or Mathematica, I end up with imaginary values. However if I solve them  numerically I obtain a nice shape $f(x,y)=0$
Due to the numerical plots I obtain I am thinking that there must be a nice (analytical) solution for these equations. How could I derive such an expression for $f(x,y)=0$?


If you do have Mathematica, use the command
GroebnerBasis[{pol1, pol2}, {x,y},{a}]
where poli are the two expressions that equal $0$. This will eliminate the variable $a$.
On WolframAlpha I got a polynomial in $x$, $y$ with fairly large coefficients.  Check this link here.
Hope there were no mistakes in the input.
$\bf{Added:}$. You will get the resultant of the two expressions considered as polynomials in $a$. The Groebner command  is just more general, can eliminate some variables ( the ones in the second group) from a system of equations.
