Question:How can I solve the complex equation z^4-2z^2+4=0?

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Discussion Overview

The discussion revolves around solving the complex equation z^4 - 2z^2 + 4 = 0. Participants explore various methods for finding the roots, including factoring, substitution, and the quadratic formula. The focus is on understanding the solutions in the context of complex numbers.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion over the solution involving complex numbers and notes their attempts at factoring and bisecting were unsuccessful.
  • Another suggests solving for z^2 first and then taking the square root of the results.
  • A different participant proposes a factorization of z^2 into two complex components and questions how to find the square roots of these factors.
  • One participant presents their results as z = ±√(1 + i√3) and z = ±√(1 - i√3), expressing uncertainty about their correctness.
  • Another participant reformulates the original equation by letting x = z^2, leading to a quadratic equation and deriving x = 1 ± √3i, and subsequently z^2 = 1 ± i√3.
  • A later reply acknowledges the previous results but questions the origin of an alternative solution provided earlier, noting that using the quadratic formula yields similar results.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of their proposed solutions. There are multiple competing views on how to approach the problem and derive the roots.

Contextual Notes

Some participants express uncertainty about the validity of their results and the methods used, indicating a lack of clarity in the steps taken to solve the equation.

JasonRox
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Complex Number, again... :(

This I'll give you the entire question and answer.

10. Solve:

[tex]z^4-2z^2+4=0[/tex]

That's all I got.

Answer:

[tex]+-1/2(\sqrt{6}+-\sqrt{2i}) (four combinations of signs).[/tex]

That is all.

I tried factoring, but I can't come up with anything. I also tried bisecting, but that is useless.

Knowing that the solution has complex numbers, I have no clue where they got this from.
 
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solve for [tex]z^{2}[/tex] then take the square root of those...
 
Ok then.

[tex]z^2=(\frac{z^2}{\sqrt{2}}-2i)(\frac{z^2}{\sqrt{2}}+2i)[/tex]

The square root of these?
 
my results are
[tex]z=+-\sqrt{1+i\sqrt{3}}[/tex]
[tex]z=+-\sqrt{1-i\sqrt{3}}[/tex]

I don't know whether they are correct or not..
 
let x= z2 then the equation z4- 2z2+ 4= 0 becomes x2- 2x+ 4= 0. That is the same as x2- 2x= -4 or
x2- 2x+ 1= (x- 1)2= -3. From that x= [itex]1+- \sqrt{3}i[/itex].
Since x= z2, [itex]z^2= 1+- i\sqrt{3}[/itex] so [itex]x= +- \sqrt{1+- i\sqrt{3}}.[/itex]
 
I can see that, but where would their solution come from?

Using the quadratic formula yield the same result as yours, too.
 

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