[Math] Complex roots forming a equilateral triangle

complex numberscomplex-analysis

Suppose we have relation
$$z^2 + az + b=0 $$ where $a$ and $b$ are real and roots of this equation $z_1$ and $z_2$ form equilateral triangle with origin then what could be relation between $a$ and $b$ ?
I simply applied quadratic formula in equation $$ z = \frac{-a \pm \sqrt{a^2 -4b}}{2} $$ now since it forms equilateral triangle with origin so $$|z_1| =|z_2|$$ applying which
$$( \frac{-a + \sqrt{a^2 -4b}}{2})^2= (\frac{-a – \sqrt{a^2 -4b}}{2})^2$$
I arrived at $$a^2 = 4b$$ but my answer is incorrect , why?

Best Answer

Hint; The triangle in the complex plane whose verticies are the origin and the points $z_{1}$ and $z_{2}$ is equalateral if and only if \begin{equation} z_{1}+z_{2} = z_{1}z_{2} \end{equation} (Further hint: The reasoning behind this is that the distance from the origin to $z_{1}$ is $\sqrt{z_{1}\bar{z_{1}}}$ etc...)

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