In an attempt to solve for stable solutions of certain non-autonomous differential equations, I arrived at the following quadratic equations $$ 0= A_n x_n^2 + B_n x_n + C_n + \sum_{m=1 | m\neq n}^N D_{n,m} x_m $$ for the variables $x_n$ and real constants $A_n, B_n, C_n, D_{nm}$ for $n,m = 1,...,N $ Can we say under which conditions on the constants, there are real solutions $(x_1,...,x_N)^T$?
For $N=1$ this becomes the standard quadratic equation $$A_1 x_1^2 + B_1 x_1 +C_1 = 0 $$ with known solution formula. The existence of solutions boils down to checking the sign of the discriminant. Solutions disappear where the determinant equals zero. Is there any such criterion for $N=2$ or larger $N$?