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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.


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: (Adobe) Portable Document Format FAU/Stuyvesant Alumni Mathematics Competition Level 1 Problems, October 2000

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