Quadratic Equation/Examples/z^4 + z^2 + 1 = 0

Example of Quadratic Equation
The quartic equation:
 * $z^4 + z^2 + 1 = 0$

has the solutions:


 * $z = \dfrac {\pm 1 \pm i \sqrt 3} 2$

Proof
Although this is a quartic in $z$, it can be solved as a quadratic in $z^2$.

$-\dfrac 1 2 \pm i \dfrac {\sqrt 3} 2$ are recognised as the complex cube roots of unity, and so:

Taking each in turn:

{{eqn | r = \map \cis {\dfrac {2 \pi} 3 + 2 k \pi | c = for $k = 0, 1$ }}

and: