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I recently encountered this problem. I wanted to solve a not so complicated equation:
Solve[K == Sqrt[(M-1)/M], M]
Mathematica returns an empty result set.
Doing so in WolframAlpha, the result is M = (1/(1-k^2), which is correct, of course.
In both programmes, I entered the exact same code.
One could think that a quite expensive Mathematica can handle at least what WolframAlpha can compute ... but here it seems not to be the case.
Any ideas? Did I do it wrong?