Finding the roots of a fourth degree polynomial

[PsychonautQQ]

Homework Statement

Find the roots of x^4 - 6x^2 - 2

Homework Equations

The Attempt at a Solution

So my first observation is that this polynomial is irreducible by Eisenstein criterion with p=2. If I substitute y=x^2 then this polynomial becomes a quadratic, and I can apply the quadratic equation to get two solutions for y. Then could I take the +/- of the square root of these solutions to get the roots in terms of x? If so, is there a simpler way to find these roots? Thanks PF!

 

[fresh_42]

Homework Statement

Find the roots of x^4 - 6x^2 - 2

Homework Equations

The Attempt at a Solution

So my first observation is that this polynomial is irreducible by Eisenstein criterion with p=2. If I substitute y=x^2 then this polynomial becomes a quadratic, and I can apply the quadratic equation to get two solutions for y. Then could I take the +/- of the square root of these solutions to get the roots in terms of x? If so, is there a simpler way to find these roots? Thanks PF!

I find this method pretty fast. The check whether it gives the correct answer has been longer than to write down the solution. And the square roots have to appear somewhere. What do you have in mind?

 

[PsychonautQQ]

I find this method pretty fast. The check whether it gives the correct answer has been longer than to write down the solution. And the square roots have to appear somewhere. What do you have in mind?

What do you mean the square roots have to appear somewhere? Yes, the check whether it gives a correct answer seems to be longer, but I could use Wolfram. I don't have any other methods in mind to solve the roots of this polynomial, I was just wondering if there was a more standard/methodical method I could use.

 

[fresh_42]

What do you mean the square roots have to appear somewhere? Yes, the check whether it gives a correct answer seems to be longer, but I could use Wolfram. I don't have any other methods in mind to solve the roots of this polynomial, I was just wondering if there was a more standard/methodical method I could use.

Well, $y^2 + px +q = 0$ let's us directly write $y_{1,2}=-\frac{p}{2}\pm \sqrt{(\frac{p}{2})^2 - q}$ which yields $x_{1,2,3,4} = \pm (\sqrt{3 \pm \sqrt{11}})$.
I meant $\sqrt{11}$ and $\sqrt{3\pm \sqrt{11}}$ have to be found somehow. So any other method has to output them and I cannot think of any faster method.

 

[PsychonautQQ]

$\sqrt{3\pm \sqrt{11}}$

Are these not all four roots we were looking for? With +/- at the beginning of course.

 

[fresh_42]

Are these not all four roots we were looking for? With +/- at the beginning of course.

Yes, and that's why I found it rather short and fast.