I would like to solve the following system of equations:
$$ \left\{ \begin{array}{r} a x^2 + b x y + c_1 x + c_2 y + d_1 = 0 ,\\ a y^2 + bxy + c_3 x + c_4 y + d_2 = 0 . \end{array} \right. $$
Here, $a,b,c_i , d_i$ are known constants and I want to solve for $x$ and $y$.
Subtracting the two equations yield $$ a (x^2 - y^2 ) + (c_1 - c_3) x + (c_2 - c_4 ) y + d_1 - d_2 = 0 $$ Though this does not give me much insight.
I think in general to solve a system of two quadratic equations in two unknowns is to do the following. For the first equation, regarding $y$ as a constant temporarily, we get $$ x = \dfrac{- (by + c_1 ) \pm \sqrt{(by + c_1)^2 - 4a (c_2 y + d_1)}}{ 2a}. $$ We can now plug this expression in the second equation and solve for $y$. This results in a fourth degree equation in $y$, which I do not know how to solve.
Any suggestions to tackle the problem would be greatly appreciated!