Question: I am looking for a generic way to solve the following system of quadratic equations:
\begin{equation} \forall 1\leq i<j\leq n, a_{ij}+b_{ij}x_{i}+c_{ij}x_{j}+d_{ij}x_{i}x_{j}=0 \end{equation}
where $a_{ij},b_{ij}, c_{ij}, d_{ij}\in \mathbb{R}$ are given parameters for all $1\leq i<j\leq n$.
My Attempt: I know that if we have $n=3$, we can potentially solve this by eliminating one variable at a time. For example, if we take
\begin{equation} a+bx_{1}+cx_{3}+dx_{1}x_{3}=0 \end{equation} \begin{equation} p+qx_{2}+rx_{3}+sx_{2}x_{3}=0 \end{equation}
we can then derive \begin{equation} (a+bx_{1})(r+sx_{2})=-(c+dx_{1})(r+sx_{2})x_{3}=(c+dx_{1})(p+qx_{2}) \end{equation}
which eliminates the variable $x_{3}$. Similarly, we can eliminate the variable $x_{2}$ as well, and we are left with a quadratic equation for $x_{1}$, which we can easily solve.
In general, the problem can be solved in the similar way, though the process becomes much more complex (it seems that we need to check a lot, lot more conditions).
Is there a compact expression (something in the feel of the determinant for 1-variable quadratic equation) that allow us to deduce if there is no real solution? Moreover, are there expressions/efficient algorithms for finding all the real solutions, if there is at least one?