<s>[INST] Find the 4 roots of the equation: $z^{4}+4=0$ Find the 4 roots of the equation: $z^{4}+4=0$
Note: I can't use the form: $e^{i\theta}$
My attempt:
Note we have: $z=(-4)^{\frac{1}{4}}$, consider $w=-4$, then $w^{1/4}=z$ this implies $w=z^4$.
Consider the polar form of $z$ and $w$:
$|w|=r=4$,
$Arg(w)=\pi=\theta$
Then, 
$$w=4\cos\pi+i\sin\pi$$
$$z=p\cos\phi+i\sin\phi$$
This implies: $w=z^4$ iff $4=p^4$ and $4\phi=\pi+2k\pi$ iff $p=4^{1/4}$ and $\phi=\frac{\pi+2k\pi}{4}$ with $k=0,1,2,3$
Then the roots are:
$$z_k=4^{1/4}(\cos{\frac{\pi+2k\pi}{4}}+i\sin{\frac{\pi+2k\pi}{4}})$$
with $k=0,1,2,3$
is correct this?
 [/INST] Consider the factorization of $z^4+4$:
$$z^4+4=z^4\color{blue}{+4z^2}+4\color{blue}{-4z^2}=(z^2+2)^2-(2z)^2=(z^2+2z+2)(z^2-2z-2)$$
Then we get $z^2+2z+2=0$ or $z^2-2z+2=0$, then finish it by using quadratic formula.
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