Question

Origin and the roots of $z^2+pz+q=0$ form an equilateral triangle if.

Answer 1

Considering the sides of the triangle to be:

$2^2=\left({\sqrt{3}}^2\right)+1^2$

$z^2+pz+q$

$=(z-k\frac{\sqrt{3}}{2}-k\frac{1}{2}i)(z-k\frac{\sqrt{3}}{2}+k\frac{1}{2}i)$

$=z^2-k\sqrt{3}z+k^2$

Comparing the equations $z^2+pz+q$ and $z^2-k\sqrt{3}z+k^2$

$q=k^2,p=-k\sqrt{3}$

$\therefore p^2=3q$