2013 Mock AIME I Problems/Problem 11
Problem
Let and be the roots of the equation , and let and be the two possible values of Find .
Solution
For some integer , let be .
Note that , so, to answer the problem, it suffices to know and .
Let . First, note the arbitrary decision in the expression . Why must be over , but not over ? From this observation, we can deduce that the aforementioned two possible values of this sum are (WLOG let this be ) and (WLOG let this be ). From these definitions and the knowledge from Vieta's Formulas that , we can now combine fractions to get the following:
Now that we have , we desire to find . Note that Vieta gives us , so, because , by substitution . By substituting our values for and into , we see the following:
Thus, we can now find our desired answer: