Find the roots of $z^6 + 3z^3 + 4 = 0$

Question

Hello I have some issues solving:

Find all $z \in C$

$z^6 + 3z^3 + 4 = 0$

How do you solve this?

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The solutions are:

1) $0.8721 \pm 0,9093i$

2) $0.3514 \pm 1.2099i$

3) $-1.2235 \pm 0.3006i$

Answer 1

$$z^3=\frac{-3\pm i\sqrt{7}}{2}=-2e^{\pm i\theta},$$ with $\tan\theta=\dfrac{\sqrt 7}3$.

Then

$$z=-\sqrt[3]2\,e^{(\pm i\theta+2k\pi)/3}, k=0,1,2.$$