# Math Help - Complex numbers

1. ## Complex numbers

Determine all complex solutions of the equation
z^4 - 4z^2 + 16 = 0
in Cartesian form.

2. Originally Posted by mamt6
Determine all complex solutions of the equation
z^4 - 4z^2 + 16 = 0
in Cartesian form.
Note that this equation is quadratic in $z^2$.

Take $w=z^2$ to turn the equation into $w^2-4w+16=0$.

Apply the quadratic formula to get $w=\frac{4\pm\sqrt{16-64}}{2}=\frac{4\pm4\sqrt{3}i}{2}=4\left(\tfrac{1}{ 2}\pm\tfrac{\sqrt{3}}{2}i\right)$.

Note that in complex polar form we have $w_1=4\left[\cos\!\left(\tfrac{\pi}{3}\right)+i\sin\!\left(\tf rac{\pi}{3}\right)\right]$ and $w_2=4\left[\cos\!\left(\tfrac{5\pi}{3}\right)+i\sin\!\left(\t frac{5\pi}{3}\right)\right]$

To get the four solutions for $z$, evaluate $z_{1,2}=w_1^{1/2}$ and $z_{3,4}=w_2^{1/2}$ by applying DeMoivre's Theorem.

Can you finish this off?