Problem:
Find , where is the sum of the absolute values of all roots of the following equation:
Solution:
The fraction on the right side of the equation can be simplified to the form
for some real numbers , and . It follows that the given equation is quadratic, and hence has at most solutions. Next, observe that any solution to
is also a solution to the original equation. This can be seen by repeatedly replacing each occurrence of in the right side of by until the equation in the problem results. Equation has two solutions,
so these must be the roots of the equation given in the problem. The sum of the absolute values of these roots is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions