This has been posted on math.stackexchange but got just one partial(insightful though) comment. I'm posting it here in a hope of getting further ideas and comments:

The problem was:

Any hope to acquire an analytic solution to such equations:

Solve: $$\sum_{j=1}^n a_{ij} x_i x_j = b_i$$

for $i=1,\ldots,n$, where $a_{ij}$'s and $b_i$'s are known constants and $x_i$'s are unknowns to be solved. Let's consider this problem in a positive setting, i.e. let's require all coefficients($a_{ij}$'s and $b_i$'s) to be positive so that the solution seems to exist. Also $a$ is symmetric, i.e. $a_{ij}=a_{ji}$. If there could be any fast numerical solution it's also useful.

Thanks a lot!