2014 AIME II Problems/Problem 5
Problem 5
Real numbers and
are roots of
, and
and
are roots of
. Find the sum of all possible values of
.
Solution
Let r, s, -r-s be the roots of p(x) (per Vieta's). Then and similarly for s. Also,
Set up a similar equation for s:
. Simplifying and adding the equations gives
Now, let's deal with the a*x. Equating the a in both equations (per Vieta)
, which eventually simplifies to
Substitution into (*) should give r = -5 and r = 1, corresponding to s = -6 and s = 9, and |b| = 330 and 90, for an answer of
.