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a= +- b if a(x-a)^2 + b(x-b)^2 = 0 has only 1 solution (2018 Auckland MO S1)

Source:

November 15, 2021
algebra

Problem Statement

For two non-zero real numbers a,ba, b , the equation, a(x−a)2+b(x−b)2=0a(x-a)^2 + b(x-b)^2 = 0 has a unique solution. Prove that a=±ba=\pm b.
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