The equation I'm trying to find the roots of is: $3x^8-12x^4 +1 =0$
The roots I want to find are the real roots, not the complex ones. Using an online calculator , it says that the roots of this equation are:
$x = \dfrac{\sqrt[4]{6-\sqrt{33}}}{\sqrt[4]{3}} \approx 0.5401828449376001$
$x = -\dfrac{\sqrt[4]{6-\sqrt{33}}}{\sqrt[4]{3}} \approx −0.5401828449376001$
$x = \dfrac{\sqrt[4]{\sqrt{33}+6}}{\sqrt[4]{3}} \approx 1.406626835288273$
and
$x = -\dfrac{\sqrt[4]{\sqrt{33}+6}}{\sqrt[4]{3}} \approx -1.406626835288273$
How do I go about finding these roots? I don't know how to use integrals and the like yet as I have not gotten to that point yet in the book I'm studying on, and so far I only know how to use derivatives and limits, so if possible I'd like it if the solutions didn't include those. If the solution has to include those, I will check those out too to see if I understand them.
thanks in advance