Question from http://www.sgforums.com/forums/2297/topics/355078
Given that 2x² + 3px - 2q and x² + q have a common factor x - a, where p, q and a are non-zero constants, show that 9p² + 16q = 0.
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Answer:
Let f(x) = 2x² + 3px - 2q and g(x) = x² + q.
By Factor Theorem, f(a) = 0 and g(a) = 0
So
2a² + 3pa - 2q = 0 -- (1)
a² + q = 0 -- (2).
From (2), a² = -q
Substitute (2)into (1):
-2q + 3pa - 2q = 0
3pa = 4q
a = 4q/3p (remember that p is non-zero, and q is non-zero)
Substitute a into (2):
(4q/3p)² = -q
16q² = -9p²q
q(16q + 9p²) = 0
But q is non-zero, hence,
9p² + 16q = 0 (shown)
