• Support PF! Buy your school textbooks, materials and every day products Here!

Solving a complex equation

  • Thread starter ƒ(x)
  • Start date
  • #1
324
0

Homework Statement



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


Homework Equations



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.
 

Answers and Replies

  • #2
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
179
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.
 
  • #3
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,227
953

Homework Statement



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

Homework Equations



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.

Added in Edit:

See Dick's method in the next post. Sweet!
 
Last edited:
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Or just write your equation as [itex](\frac{z+1}{z})^5=1[/itex]. That tells you 1+1/z is a fifth root of unity. It's pretty easy from there.
 

Related Threads for: Solving a complex equation

  • Last Post
Replies
4
Views
580
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
11
Views
8K
  • Last Post
Replies
3
Views
519
  • Last Post
Replies
1
Views
2K
Replies
8
Views
1K
Replies
5
Views
1K
Replies
5
Views
1K
Top