<s>[INST] How to find the roots of $x^4 +1$ I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ is a positive term as the equation above.
Thanks
 [/INST] We have:
$$\begin{align}x^4+1 =0 \\
x^4+1+2x^2-2x^2 =0 \\
(x^2+1)^2-(\sqrt{2}x)^2 =0 \\
(x^2+1+\sqrt{2}x)(x^2+1-\sqrt{2}x) =0 \end{align}$$
Then on solving these factors by quadratic formula, we will have:
$$x= \frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}} , \frac{-1}{\sqrt{2}}-\frac{i}{\sqrt{2}}, \frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}, \frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}$$
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