How to solve this uni-variate equation? I am given that:
$$\sum_{i=1}^{n}{w_{i}}=1\\$$
and that a set of numbers $e_i$, where $i$ can range from $1$ to $n$.
Now I need to find a number $u$, such that
$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$$
My questions are: are there systematic way of finding all possible solutions $u?$
And is the number of solution related to $n?$ 
I am thinking of maybe for $n=2$, the number of solutions $u$ is $1?$
$u$ is unconstrained... all the rest are given...
and yes, $w_i >$ or = $0$ for all $i$...
And for general $n$, the number of solutions $u$ is $n-1$?
Thanks a lot!
[Edit]
Now I need to find a number $u$, such that
$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$$
And I am looking for real numbers $u$...
And after finding all these roots $u$'s,
I would like to compare all of the following:
$$\sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}/e_{i}^{2}}$$
and find one of the roots u* which maximizes the above expression?
Any possible shortcuts?
Thanks
 A: If any of the $w_i$ are zero, they don't affect either sum, so we may assume $w_i\gt0$ for all $i$. 
As $u\to\infty$, the sum approaches zero. If $u$ is just a hair bigger than the biggest $e_i$, then the sum is enormous. Between the biggest $e_i$ and $\infty$, the sum is decreasing. It follows that there is a unique solution $u$ strictly between the biggest $e_i$ and $\infty$. 
Two questions remain: how to find that $u$, and whether there are any solutions less than the biggest $e_i$. 
Both of these seem difficult. Clearing denominators yields an equation of degree $2n$ in $u$, so I think only numerical methods (e.g., Newton's Method) apply. And what happens between the $e_i$ is not obvious to me. 
A: I will rewrite your equations to be more readable:
$$\sum_{i=1}^{n}{w_{i}}=1\\ \sum_{i=1}^{n}{\left(\frac{w_{i}}{u-e_{i}}\right)^{2}}=1$$
I thought about least squares solution, but it seems the problem is quite different...
A: This question reminds me of the 1-D problem of computing the electrostatic force between a test charge $q$ and a collection of charges $q_i$, where $i = 1,...,n$.  The electrostatic force is
$$
F(r) = \frac{q}{4 \pi \epsilon_0} \sum_{i=1}^n \frac{q_i}{(r-r_i)^2} \; .
$$
