The real part of each root of the equation (z+1)^6 + z^6=0 is

Any solution to the equation zⁿ = (z+1)ⁿ satisfies |z| = |z+1| i.e. the point is equidistant from (0,0) and (-1,0).

This means it lies on the perpendicular bisector of the line joining (0,0) and (-1,0) i.e. x = -1/2 or 2x+1 = 0.

Regards

Arun