<s>[INST] Solve $z^4+1=0$ algebraically I know the result and how to solve it using trigonometry and De Moivre. 
However, given that the complex number $z$ can be rewritten as $a+bi$, how can I solve it algebraically?
 [/INST] $$ z^4 + 1 = 0 \Rightarrow z^4 + 2z^2 + 1 - 2z^2 = 0 $$
$$ \Rightarrow (z^2 + 1)^2 -(\sqrt{2}z)^2 = 0 $$
$$ \Rightarrow (z^2 - \sqrt{2} z + 1)( z^2 + \sqrt{2}z + 1 ) = 0 $$
use $$ ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
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