Understanding the Roots of a Quadratic Equation

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Homework Help Overview

The discussion revolves around understanding the roots of a quadratic equation, specifically focusing on the roots derived from a quartic polynomial, \(x^4 + 4\). Participants are exploring the nature of these roots, particularly the significance of the powers involved in the solutions provided by Wolfram Alpha.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the origin of the 3/4 power in the roots and discussing the relationship between the roots of unity and the solutions. There are suggestions to factor the polynomial and inquiries about the relevance of roots of unity in this context.

Discussion Status

Some participants have offered guidance regarding the roots of unity and suggested factoring the polynomial as a potential approach. There is an acknowledgment of the original poster's confusion and a willingness to explore the problem further.

Contextual Notes

The original poster indicates that this is course work rather than homework, suggesting a different level of expectation regarding the assistance sought. There is also mention of a resource in the form of a textbook that discusses roots of unity, which may provide additional context for the problem.

nmsurobert
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Homework Statement


I'm looking for an explanation to something. I've attached a picture of the solution wolfram alpha is giving me.
I understand the first two zeros, +- (-1)^(1/4)*sqrt(2).
But i don't understand the other two zeros with the 3/4 power. Where does that power come from?
image.png


This isn't homework but it is course work. I thought I try some of the other problems out before diving into the actual homework and I'm already stuck lol.

Homework Equations

The Attempt at a Solution

 
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Just a few questions: Why don't you supply the link to Wolfram instead? This would definitely result in a better resolution.
Anyway, have you tried to factorize ##x^4 + 4## or to put it another way: what do you know about the roots of unity?
 
nmsurobert said:

Homework Statement


I'm looking for an explanation to something. I've attached a picture of the solution wolfram alpha is giving me.
I understand the first two zeros, +- (-1)^(1/4)*sqrt(2).
But i don't understand the other two zeros with the 3/4 power. Where does that power come from?
View attachment 105722

This isn't homework but it is course work. I thought I try some of the other problems out before diving into the actual homework and I'm already stuck lol.

Homework Equations

The Attempt at a Solution

The wolfram page mentions that these roots are multiples of the four fourth roots of unity (1). The fourth roots of 1 are i, -1, -1, and 1. They are equally spaced around the unit circle.
 
fresh_42 said:
Just a few questions: Why don't you supply the link to Wolfram instead? This would definitely result in a better resolution.
Anyway, have you tried to factorize ##x^4 + 4## or to put it another way: what do you know about the roots of unity?
Sorry. I'm on my phone and took a quick screen shot.
I haven't tried to factorize it yet. I'll give that a shot. I know there is some symmetry in factoring problems like that.
I don't know anything about roots of unity.
I figured there was some relation between the first two solutions being 1/4 power and the other two being 3/4 power. I'm assuming "roots of unity" plays a part there.
 
Thanks guys. The book has two pages on roots of unity that help with this problem.
 

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