Solving this set of quadratic equations

Question

I have a set of quadratic equations of the form:

\begin{equation*} 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0 \\ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0 \\ \vdots \\ 2S_nq_n(q_0S_0 + q_1S_1 + ...... + q_nS_n)+k_n = 0 \\ \end{equation*}

We have $n$ equations and $n$ variables $q_0,q_1,q_2.....q_n$. The $k$ and $S$ values are constants.

How can I analytically reach a solution for this system?

Answer 1

Hint:

Set $X = q_0S_0 + q_1S_1 + \dots + q_n S_n$ and add up all the equations.