# Solving a complex equation

## Homework Statement

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

None

## 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.

Related Calculus and Beyond Homework Help News on Phys.org
jbunniii
Homework Helper
Gold Member
Solve (z+1)^5 = z^5
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.

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

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

None

## 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.
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.