Complex number equation of 10th degree

Question

I'm asked to show, "without solving the equation", that given:

$(z-1)^{10}=z^{10}$,

then $\Re(z)= 1/2$

To be honest I don't even know how to solve the equation in the first place...

Answer 1

Hint: $\displaystyle(z-1)^{10}=z^{10}\implies\lvert z-1\rvert^{10}=\lvert z\rvert^{10}\iff\lvert z-1\rvert=\lvert z\rvert$.