Mock AIME II 2012 Problems/Problem 12
Let . Assume the value of has three real solutions . If , where and are relatively prime positive integers, find .
Let . Then and . From this, we have the system
Substituting the first equation into the second, we obtain
Plugging this into the third equation yields .
Thus, . Note that our three real roots multiply to . However, since , we need to multiply by , so our is
We need . Using vieta’s and making sure we count for each factor of we divided off, we have .
Our answer is , thus .