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

## Example of Complex Algebra

The roots of the equation:

$z^2 \paren {1 - z^2} = 16$

are:

$\pm \dfrac 3 2 \pm \dfrac {\sqrt 7} 2 i$

## Proof

Let $w := z^2$.

Then:

 $\ds w \paren {1 - w}$ $=$ $\ds 16$ $\ds \leadsto \ \$ $\ds w^2 - w + 16$ $=$ $\ds 0$ $\ds \leadsto \ \$ $\ds w$ $=$ $\ds \dfrac {1 \pm \sqrt {1^2 - 4 \times 1 \times 16} } 2$ Quadratic Formula $\ds$ $=$ $\ds \dfrac {1 \pm 3 \sqrt {-7} } 2$ $\ds$ $=$ $\ds \dfrac 1 2 \pm \dfrac 3 2 \sqrt 7 i$

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

Then:

 $\ds \paren {p + i q}^2$ $=$ $\ds p^2 - q^2 + 2 p q i$ $\dfrac 1 2 \pm \dfrac 3 2 \sqrt 7 i$ $\ds \leadsto \ \$ $\ds p^2 - q^2$ $=$ $\ds \dfrac 1 2$ $\ds 2 p q$ $=$ $\ds \pm \dfrac 3 2 \sqrt 7$ $\ds \leadsto \ \$ $\ds q$ $=$ $\ds \dfrac {1 \pm 3 \sqrt {-7} } {4 p}$ $\ds \leadsto \ \$ $\ds p^2 - \paren {\dfrac {1 \pm 3 \sqrt {-7} } {4 p} }^2$ $=$ $\ds 16$

which leads after unpleasant algebra to:

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

$\blacksquare$