Complex Algebra/Examples/z^2 (1 - z^2) = 16/Proof 2

Proof
Let $w := z^2$.

Then:

Let $\paren {p + i q}^2 = \dfrac 1 2 \pm \dfrac 3 2 \sqrt 7 i$

Then:

which leads after unpleasant algebra to:


 * $p + i q = \pm \dfrac 3 2 \pm \dfrac {\sqrt 7} 2 i$

Note:


 * Alternatively, the formulae in Square Root of Complex Number in Cartesian Form could be used to calculate $\sqrt {\dfrac 1 2 \pm \dfrac 3 2 \sqrt 7 i}$ directly thereby avoiding having to compare the real and imaginary parts of $\paren {p + i q}^2$ and the ensuing unpleasant algebra.