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

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

has the solutions:


 * $z = \begin{cases} \dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\

\dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \end{cases}$

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

From the Quadratic Formula:

It remains to solve for $z$.

Let $z = x + i y$.

We have:

Thus we have:


 * $z = \begin{cases} \dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\

\dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 + i \dfrac {\sqrt 2} 2 \\ -\dfrac {\sqrt 6} 2 - i \dfrac {\sqrt 2} 2 \end{cases}$