Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + ... + 1/(x+a_n) = 1/x$ are all real Prove that the roots of the equation:
$$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$
are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.

This is a fairy intuitive way of approaching it.
Consider 2 functions, $$y=\frac{1}{x}$$ $$y=\sum_{i=1}^n \frac{1}{x+a_i}$$
Since $a_i>0$, graph the 2 functions. Graph
Without loss of generality, assume that $\prod_{i=1}^n a_i\neq0 $
Trivially, the original equation is $n-1$th-degree polynomial equation, so by the Fundamental Theorem of Algebra, it can have at most $n-1$ roots.
From the graph, we can identify $n-1$ intersection points.
Therefore, all roots are real.
