Consider the variables $\mathbf{x}\in\mathbb{R}^n$ and the known coefficients $\mathbf{A}_i \in \mathbb{R}^{n\times n}, \mathbf{A}^T = \mathbf{A}, \mathbf{b}_i \in \mathbb{R}^n,$ and $c_i \in \mathbb{R}$ for $i=1,2,\cdots, n$. They satisfy an $n$-equation system given by: $$ \mathbf{x}^T\mathbf{A}_i \mathbf{x} + \mathbf{b}_i^T \mathbf{x} + c_i = 0, \quad i=1, 2,\cdots,n. $$
Question: Could someone please give me some hints about the existence of a solution for this particular equation system? This question is extended from the topic discussed in this post.
Constraints: In my application, each $\mathbf{A}_i$ is a symmetric positive semi-definite matrix with rank at most $3$ and ($n > 3$). Also, $\mathbf{b}_i$ is in column space of $\mathbf{A}_i$ (, or row space of $\mathbf{A}_i$ since $\mathbf{A}_i$ is symmetric). However, I would like to know what general constraints should be added to ensure the existence of a solution.
Initial thoughts: To guarantee the existence of a solution for this system, one should guarantee the solution existence for each equation. To determine this, we can quickly check the definiteness of $\mathbf{A}_i$ and examine the minimum or maximum of each LHS. The process for checking this is explained in detail in this post. However, this is not sufficient to determine the existence of a solution for the entire system, as the solution for the system requires the intersection of all solutions of each equation.
Related post: I could only locate one closely related post on MathOverflow. However, the answer only supports $\mathbf{B}=[\mathbf{b}_1^T, \mathbf{b}_1^T,\cdots, \mathbf{b}_n^T]$ should be nonsingular M-matrix. In my application, $\mathbf{B}$ does not have such constraints but sometimes could have a solution via the Newton method.