# Roots of the system of quadratic equations

For a set of $$m$$ positive semi-definite $$dtimes d$$ matrices $$Q_j$$ I have the following system of equations over column-vector $$vec{x}$$:
$$q_j = vec{x}^T Q_j vec{x}, quad j=1,dots,m$$
with some known non-negative $$q_j$$.

1. Are there effective numerical solvers for this type of systems?
2. For a given solution $$vec{y}$$ there also exists the solution $$-vec{y}$$. Is there a way to prove that there are no other solutions exists for the specific problem?