CMO 2001 Problem 1

Randy: "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?''

Rachel: "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.''

Randy: "That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn't be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.''

Rachel: "Interesting. Now figure out how old I am.''

Randy: "Instead, I will guess your age and substitute it for $x$ in your quadratic equation $\dots$ darn, that gives me $-55$ , and not $0$ .''

Rachel: "Oh, leave me alone!''

(1) Prove that Jimmy is two years old.

(2) Determine Rachel's age.