Let Z_1 and Z_2 be two roots of Z^2+aZ+b=0, Z complex.

Question

Let $Z_1$ and $Z_2$ be two roots of $Z^2+aZ+b=0$, $Z$ complex. If the origin, $Z_1$, $Z_2$ form an equilateral triangle, then

Accepted Answer

$Z_1+Z_2=-a$, $Z_1Z_2=b$. Equilateral condition: $Z_1^2+Z_2^2=Z_1Z_2 \implies (Z_1+Z_2)^2=3Z_1Z_2 \implies a^2=3b.$

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