Secondary student here (or 9th Grade), so sorry if I find it hard to understand, as we have not been taught this and probably won't for a few years to come.
I have come across a question where I have been given the polynomial $z^4-6z^3+24z^2-18z+63$ where $z^2 + 3 = 0$ is a known root. Therefore I found the roots to be $\pm i\sqrt{3}$ and after factorisation $3 \pm 2i\sqrt{3}$.
I gave this a brief sketch on the complex plane, but how would I find the exact equation of this circle?
Sorry if this question is a common one, I couldn't seem to find out how anywhere, and any help would be greatly appreciated.
Maybe this sketch will give you more ideas. You can try to find a complex $z =a+i \,b$ such as for all your roots $r_1,r_2,r_3,r_4$ :
$|z-r_1| =|z-r_2| = |z-r_3| = |z-r_4|$
I don't know if it helps you. If you succeed to do so then the center of the circle will be $(a,b)$ and the radius $R = |z-r_1|.$