AIME Qualifiers Compete

Two Owls qualified for the American Invitational Math Exam (AIME). Freshman Dannie Dong and seventh-grader James Jin compete this month to qualify for the even-more selective group competing in the United States of America Junior Mathematical Olympiad in April.
A sample AIME question follows; send your answers to Dr. Steve Gadbois at your leisure:
 
Let p<q<r be the roots (solutions) of the equation 2014^(1/2) x^3 - 4029 x^2 + 2 = 0. Find the value of q(p+r).
 
Two hints:
  • First, note that 4029=1+2*2014.
  • Second, don't immediately try to find the individual values of p, q, or r.
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