Understanding the Roots of a Quadratic Equation

[nmsurobert]

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?

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

 

[fresh_42]

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?

 

[Mark44]

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.

 

[nmsurobert]

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.

 

[nmsurobert]

Thanks guys. The book has two pages on roots of unity that help with this problem.