We have $A\_i , B\_i , C\_i , D\_i , E\_i ,F\_i, \ (i=1, 2) $.


We want to find $ (u,v) \in \mathbb{R}^2$ satisfying


\begin{equation}
A\_1 u^2 + B\_1 uv + C\_1 v^2 + D\_1 u + E\_1 v +F\_1 =0 \\
A\_2 u^2 + B\_2 uv + C\_2 v^2 + D\_2 u + E\_2 v +F\_2 =0.
\end{equation}


Is there a general method to find exact solution (not an approximate solution)?



Cross section point of two conics curves
Take the resultant of the two equations with respect to one of the variables, say $v$. You'll get a quartic polynomial in $u$ whose coefficients are (complicated) polynomials in $A\_1,\ldots,F\_2$, but if $A\_1,\ldots,F\_2$ are real numbers, as you seem to suggest, then you'll get a quartic in $u$ with real coefficients. Then you can use the formula giving the exact solution to a quartic equation to find the four possible values of $u$. Similarly, you can find the four values of $v$. Then it's just a matter of matching which $u$ goes with which $v$. (You say your want to find the solutions in $\mathbb R$, but of course, sometimes there will only be solutions in $\mathbb C$.)


