Let's consider a system of $n$ equations and $n$ real unknowns $(x_1, \dots, x_n)$:
$$\sum_{k=1}^n A_{ik} x_k = B_i, \qquad \text{with } i\in\{1,\dots,n\}$$
The solution of which exists as long as the determinant $A = \det(A_{ij})$ is not zero. However, the seemingly innocent second-order system given by:
$$\sum_{j,k} A_{ijk} x_j x_k + \sum_j B_{ij}x_j + C_ i = 0, \qquad \text{with } i\in\{1,\dots,n\}$$
It seems to have no general solution, I wonder two things:
- There is something similar to the discriminant $\Delta$ of a quadratic equation that tells me something about the set of solutions (in the case of an unknown the discriminant informs me whether there are one, two, or no real solutions).
- What is the simple reason why there is no simple procedure to find solutions to a system of quadratic equations?