Help with finding all real solutions

[akp]

Homework Statement

x^4 - 8x^2 + 2 = 0

Homework Equations

quadratic formula?

The Attempt at a Solution

i would've tried using the quadratic formula, but I am not sure if this would work with that seeing as how it's not ax^2 + bx + c

 

[fghtffyrdmns]

You will need to use the factor/remainder theorem.

 

[symbolipoint]

The example equation is in quadratic form, or something like it. First, you may use a substitution, something like, t=x^2. This can give you like, this:

t^2 - 8t + 2 = 0

Now, you can solve THAT one and you may obtain two solutions, but those solutions are for t. Now, solve each of those solutions for x (remember to first replace t with x^2 )

 

[Elucidus]

The polynomial fortunately is quadratic, but in x².

If u = x² then you have x⁴ -8x² + 2 = u² - 8u + 2 = 0.

You can use the quadratic formula to solve for x² keeping in mind that any negative roots from the formula must be discarded since x² can never be negative.

--Elucidus

 

[lurflurf]

...keeping in mind that any negative roots from the formula must be discarded since x² can never be negative.

--Elucidus

We have this field now called complex numbers that has negitive squares.
i^2=-1 for example

 

[Elucidus]

We have this field now called complex numbers that has negitive squares.
i^2=-1 for example

The thread title is "help with finding all real solutions" so I figured that complex solutions weren't needed.

--Elucidus

 

[lurflurf]

The thread title is "help with finding all real solutions" so I figured that complex solutions weren't needed.

--Elucidus

Good point. I like to keep track of the complex roots to assist in tracking the real roots. The fundamental theorem of a algebra impies each complex root means one less real root to find.