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Q.
The origin and the roots of the equation $z ^2+ pz + q =0$ form an equilateral triangle if -
Complex Numbers and Quadratic Equations
Solution:
$z^2+p z+q=0$
$z_1+z_2=-p, \,\,\,\, z_1 z_2=q$
If $z_1, z_2, z_3$ are the vertices of an equilateral triangle then $z_1^2+z_2^2+z_3^2=z_1 z_2+z_2 z_3+z_3 z_1$
If $z_3$ is origin then $z_1^2+z_2^2=z_1 z_2$
$\Rightarrow\left(z_1+z_2\right)^2=3 z_1 z_2 \Rightarrow p^2=3 q$