How to find the solutions $x $ of the following equation: $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.

For the case $k=2$, I get the solutions $\pm \sqrt{n_1n_2}$. I was trying to use Vieta's formulas to simplify the expression, but I am unable to do make any progress. Kindly share your thoughts. Thank you.

1. I was trying induction on $k$ but didn't lead anywhere.
2. I tried to represent it as some indefinite integral. But no success.

Also, is there any theoretical significance of these polynomials? Kindly share some references. Thanks again.