Solving general multivariable quadratic equations Consider the variables $\mathbf{x}\in\mathbb{R}^n$ and the known coefficients $\mathbf{A}_i \in \mathbb{R}^{n\times n}, \mathbf{b}_i \in \mathbb{R}^n,$ and $c_i \in \mathbb{R}$ for $i=1,2,\cdots, n$. They satisfy an $n$-equation system given by:
$$
\mathbf{x}^T\mathbf{A}_i \mathbf{x} + \mathbf{b}_i^T \mathbf{x} + c_i = 0, \quad i=1, 2,\cdots,n.
$$
Can anyone suggest a general solver that can be used for this type of system? I'm not well-versed in algebra and would appreciate it if you could provide me with some method names that I can use.
 A: $
\def\e{\varepsilon}
\def\p{\partial}
\def\LR#1{\left(#1\right)}
\def\BR#1{{\large\tt[}#1{\large\tt]}}
\def\op#1{\operatorname{#1}}
\def\trace#1{\op{Tr}\LR{#1}}
\def\qiq{\quad\implies\quad}
\def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red}{#1}}
\def\Sk{\sum_{k=1}^n}
$Let $\{\e_k\}$ denote the standard basis vectors and define the following variables
$$\eqalign{
A_k^s &= \tfrac12\LR{A_k+A_k^T}
  \qquad\{\,{\rm symmetric\;part\;of}\:A_k\} \\
f_k &= {x^TA_k^s x \,+\, b_k^Tx \,+\, c_k} \\
y_k &= {2A_k^s x} \\
Y &= \BR{y_1\;y_2\;\cdots\;y_n}^T \;\equiv\; \Sk\e_k\,y_k^T \\
B &= \BR{b_1\;b_2\;\cdots\;b_n}^T \\
}$$
so that $\{b_k,y_k\}$ are the $k^{th}\,$ rows of $\,\{B,Y\}$ respectively.
Calculate the differential of $f_k$
$$\eqalign{
df_k &= \LR{2A_k^s x + b_k}^T dx \;\equiv\; \LR{y_k+b_k}^T dx \\
}$$
Summing with the $\{\e_k\}$ yields a system of non-linear equations (NLEs) and its Jacobian
$$\eqalign{
f &= \Sk \e_k\,f_k = 0 \quad &\{ {\rm NLEs} \} \\
df &= \Sk \e_k\,df_k \\&= \LR{Y+B}\cdot dx \\
\grad{f}{x} &= \LR{Y+B} \;=\; J \quad &\{ {\rm Jacobian} \} \\
}$$
Now you can solve the problem using Newton's Method
$$\large\eqalign{
x_+ &= x - J^{-1}f \\
}$$
