Solution to puzzle 103 (Quadratic roots)

Find a necessary and sufficient condition for one of the roots of x² + ax + b = 0 to be the square of the other root.

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Let the roots be r, s.  One root is the square of the other if, and only if

0	= (r − s2)(s − r2)
	= rs + r2s2 − (r3 + s3)
	= rs + r2s2 − (r + s)3 + 3rs(r + s)
	= b + b2 + a3 − 3ab  (since r + s = −a, rs = b.)

Source: FAU/Stuyvesant Alumni Mathematics Competition Level 1 Problems, October 2000

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