What are all the possible values of $x$ if $(x^2-5x+5)^{x^2-7x+12}= 1$. I have already found three answers: 4, 3, and 1, however, apparently there are more possibilities. I don't know how to figure this out so it would be extremely appreciated if someone found the other possibilities and showed me how to do it.
2026-03-26 16:05:51.1774541151
Solve $(x^2-5x+5)^{x^2-7x+12}= 1$.
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Hint:
Solving this, we get $x^2-5x+6=0\rightarrow x=2,3.$
We check that the exponent is even.
If $x=3,$ then $(x-3)(x-4)=0.$
If $x=2,$ then $(-1)(-2)=2.$
Either way, it is even, so both solutions are valid.
All solutions are $ 1,2,3,4$.