Question

Find all solutions of (z+1)^5 = z^5.
I am confused about something

Answer 1

I am looking at this solution here
http://math.stackexchange.com/a/928749

Answer 2

How did they derive the following.

1. (z+1)^5 = z^5 implies | z + 1 | ^2 = |z|^2

2. | z + 1 | ^2 = |z|^2 implies Re(z) = - 1/2

Answer 3

1. take the fifth root ? then z +1 = z
then |z + 1| = | z |

2. not sure

Answer 4

2. the distance from -1 is the same as the distance from 0.
therefore the real part must be -1/2 .

Answer 5

| z - (-1) | = | z - 0 |