How Do You Solve w=0.5(z+1/z) for Complex w?

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Discussion Overview

The discussion centers on solving the equation w=0.5(z+1/z) for complex values of w, particularly focusing on proving that there are exactly two solutions for w≠±1. The scope includes mathematical reasoning and exploration of complex numbers.

Discussion Character

  • Exploratory, Mathematical reasoning, Homework-related

Main Points Raised

  • One participant expresses confusion about proving the existence of two solutions for the given equation and seeks guidance.
  • Another participant suggests multiplying both sides of the equation by 2z as a potential step in the solution process.
  • A participant proposes that finding the solutions directly is the best approach, referencing the quadratic formula as applicable even for complex coefficients.
  • There is a mention of the quadratic formula and its use in solving for z, with a note that substituting values for w or z could help in determining the other variable.
  • One participant acknowledges their uncertainty about finding the square root of a complex number but suggests writing the solution in radical form.
  • A new participant shares their first post, indicating that they are still learning and unsure about the solution process.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method to prove the existence of two solutions, and multiple approaches are suggested without agreement on a definitive solution.

Contextual Notes

There are unresolved aspects regarding the application of the quadratic formula and the handling of complex numbers, as well as the dependence on specific substitutions for w or z.

Macarenses
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How does one prove that for w≠+-1 , w a complex number, there are exactly 2 solutions to the equation w=0.5(z+1/z)? I'm at a total loss here. Could someone clue me in on this one?
 
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Macarenses said:
How does one prove that for w≠+-1 , w a complex number, there are exactly 2 solutions to the equation w=0.5(z+1/z)? I'm at a total loss here. Could someone clue me in on this one?
What if you multiplied each side by 2z?
 
Last edited:
The best way to prove there are two solutions is to find them! That is what ramsey2879 is suggesting.
 
HallsofIvy said:
The best way to prove there are two solutions is to find them! That is what ramsey2879 is suggesting.
I do believe that the quadratic formula will work for equations with complex coefficients even if it is not clear (at least to me) how to find the square root of a complex number. Just write the solution out in the radical form simplified as far as posible.
 
Do a internet search: complex numbers solve square root
 
w=0.5(z+1/z)
w=0.5z+0.5/z
wz=0.5z^2+0.5
0.5z^2+0.5-wz=0

If you could substitute a value of either w or z you could work out the other using Quadratic Formula

x = (-b +- sqrt(b^2-4ac))/2a

Other than this, I can't see how you solve it since you have 2 unknowns. Either way you have a squared value for z so z has 2 solutions.

Sorry if I was no help, this is my first post here as I just joined today. :)
 

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