10th grade algebra contest math problem

Find the all real solutions to $$x(x-1)^{x-1/x}=1$$

This question was asked to 10th grade students by a teacher who was preparing questions for the contest mathematics.The teacher said she would share the original answer to the question soon, but I can't wait. I wanted to ask here.

WA could not find the exact answer and gives an approximate solution as $$x\approx 1.61803398874989...$$ The equation seems really difficult and the way to approach the solution is not clear. There are very few elementary ways to try. Graphically the equation has clearly one real root.

How do we solve that strange equation?

• The result seems to be the golden ratio. Commented Sep 9, 2023 at 20:08
• Golden Ratio ? Fantastic !! Commented Sep 9, 2023 at 20:10
• The golden ratio $\phi$ satisfies the equation $\phi^2-\phi-1=0$, hence $\phi-\frac{1}{\phi}=1$ and $\phi^2-\phi=1$, therefore $\phi$ is a solution to the equation $x(x-1)^{x-1/x}=1$: $$\phi(\phi-1)^{\phi-1/\phi}=\phi(\phi-1)^1=\phi^2-\phi=1.$$ Commented Sep 9, 2023 at 20:32

I'm not sure if there is an intuitive way to see this other than noting that the golden ratio solves $$x^2 - x-1 = 0$$ and factoring
\begin{align} &x(x-1)^{x-1/x} \\ &=x(x-1)^{(x^2-1-x+x)/x} \\ &=x(x-1)\cdot (x-1)^{\frac{x^2-x-1}{x}} \\ &=(1 + [x^2-x-1])\cdot (x-1)^{\frac{x^2-x-1}{x}} \end{align}
• Can you use $1/\phi = \phi - 1$?