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About the equation

I posted the problem about the equation \(x^{2x-1}=2\) indicating to find a rational root. But that is to determine a rational root if it exists, or prove that the equation does not have a rational root. When solving the equation, we can assume that \(x=\frac{p}{q}\) where \(p\) and \(q\) are integers, but that approach is considering that the root is a rational number.

Now in this note, I open a discussion about the problem to solve the equation without considering that the root is rational. Considering that the root is rational we can to find it, but, (assuming): what would happen for example, if a rational root does not exist? We can show that, but then, because of to assume that \(x=\frac{p}{q}\) where \(p\) and \(q\) are integers did not work, how would we solve the equation?

That is to say, let's solve the equation algebraically without assume that \(x=\frac{p}{q}\) where \(p\) and \(q\) are integers.

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