What can we say if the roots of a cubic are in arithmetic progression?
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Ref: R6276

Question

Given that the roots of the equation \(x^3+px^2+qx+r=0\) are three consecutive terms of an arithmetic progression, show that\[ 2p^3 + 27r = 9pq.\]
Given that the roots of the equation \(x^3+px^2+qx+r=0\) are three consecutive terms of a geometric progression, find a condition that \(p\), \(q\) and \(r\) must satisfy.