Using the fact that $|s|^2 = ss^*$ (where $s$ is a complex number, and $s^*$ it's conjugate) or other methods, how can I find the root of the following independent equations:
$$ z^4 - 1 = 0 \\ z^3 + 6z +20 = 0 $$
It would be nice if you could explain the general reasoning, as opposed to showing how it works in this particular example so that the SE community and I can tackle future problems like these.
The question is slightly unclear to me. However, the general method is the following:
We know how to find solutions of quadratic equations; using the quadratic formula. These are not quadratic equations, so lets try to reduce them to something of the form of a quadratic equation (or even simpler, if we're lucky).
$$0=z^4 - 1 = (z^2 - 1)(z^2 + 1)$$
We now have two factors, both of which are quadratic. We can use the quadratic formula on each of them in turn to find solutions.
In your second example,
$$z^3 + 6z + 20 = 0 $$
we have a little more trouble, as we can't see so clearly how to break this down into it's constituent factors. A good first step is to guess at some solutions. After a few tries, we figure out that
$z=-2$
is a solution. This corresponds to a factor of $(z+2)$. So we know, now, that
$$z^3 + 6z + 20 = (z+2)*B$$
Where B is the other factor. So what is B? Well, we need a thing which multiplies with the $(z+2)$ factor to give our original expression. You can do this any way you like to find out what B is, but I prefer the following:
$$z^3 + 6z + 20 = (z+2)*(z^2+kx+10)$$
Where $k$ is unknown. We know that we want the first and last terms to be $z^2$ and $10$, because they need to multiply to give $z^3$ and $20$, respectively, in the original expression.
Expand the left hand side out and find out what $k$ is. You then have a quadratic which you can solve, using the formula.
