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Relative Powers

Stage: 4 Short Challenge Level: Challenge Level:1

Let the number be $x$.

Then $x^3 = 9x$, so either $x=0$ or $x^2 = 9$, giving $x = 3$ or $x = -3$.

Of these three values of $x$, only $-3$ is also a solution of the equation $x^2 = x + 12$.

This problem is taken from the UKMT Mathematical Challenges.