Solution for system of quadratic equations Can anyone provide a straightforward solution to the following equation:
$\vec{y}=M\vec{x}+N\vec{x^2}$
where $\vec{x^2}$ is a column vector with each component being the squared value of the components in $\vec{x}$.  Also, $M$ and $N$ are constant coefficient matrices?  My current problem is a 3-dimensional system, so I know there are 8 solutions, but I cannot come up with an elegant process.  I could brute-force it, but that gets very messy very quickly.  I've seen similar problems where dealing with a scalar equation, but nothing quite in this form.  Thanks in advance for any help!
 A: This might be a good way to start: consider the system
$$
y = M x_1 + N x_2
$$
where $x_1$ and $x_2$ are independently chosen.  This is now a system of $2n$ linear equations (rather than $n$ non-linear ones).  That is, we can think of this as the system
$$
\pmatrix{M&N}\pmatrix{x_1\\x_2} = y
$$
Once we've solved this, we can make the additional constraint $x_2 = (x_1)^2$ on the solution set.

In particular: let's assume that $M$ and $N$ have full rank, and that our system is $3$-dimensional.  By the above method, we'll find the solution looks something like
$$
\pmatrix{x_1\\x_2} = \pmatrix{x_{10}\\x_{20}} 
+ a_1 \pmatrix{x_{11}\\x_{21}}
+ a_2 \pmatrix{x_{12}\\x_{22}}
+ a_3 \pmatrix{x_{13}\\x_{23}}
$$
where $a_i$ are undetermined scalars and $x_{ij}$ are fixed vectors found by the usual solution process.
With the constraint in place, we now have the system on the three variables $a_1,a_2,a_3$ given by
$$
(x_{10} + a_1 x_{11} + a_2 x_{12} + a_3 x_{13})^2= 
x_{20} + a_2 x_{21} + a_2 x_{22} + a_3 x_{23}
$$
