# Homework Help: Solving a complex equation

1. Sep 20, 2012

### ƒ(x)

1. The problem statement, all variables and given/known data

Solve (z+1)^5 = z^5

2. Relevant equations

None

3. The attempt at a solution

z^5 + 5z^4 + 10z^3 + 10z^2 + 5z + 1 = z^5
5z^4 + 10z^3 + 10z^2 + 5z + 1 = 0
5z^3(z + 2) + 5z(2z + 1) = -1

I'm not quite sure how to go about solving this. Expanding, canceling terms, and then factoring doesn't get me anywhere.

2. Sep 20, 2012

### jbunniii

It's clear that there are no real roots: consider the graphs of (x+1)^5 and x^5 for real x. Are you looking for an exact solution or a numerical approximation? For an exact solution, you'll have to solve a quartic equation, which can be done but it's fairly ugly.

3. Sep 20, 2012

### SammyS

Staff Emeritus
Play with it.

5z4 + 10z3 + 10z2 + 5z + 1
=5(z4 + 2z3 + 2z2 + z) + 1

=5(z4 + 2z3 + z2 + z2 + z) + 1

=5( (z2 + z)2 + (z2 + z) ) + 1

=5 (z2 + z)2 + 5 (z2 + z) + 1 ​

Let u = z2 + z .

You have a quadratic equation in u .

Solve for u, then solve u = z2 + z for z.

Or just write your equation as $(\frac{z+1}{z})^5=1$. That tells you 1+1/z is a fifth root of unity. It's pretty easy from there.