Multiple choice

$z_1, z_2$ are the roots of the equation $z^2+az+b=0$. Let $z_1$, $z_2$ and the origin be the vertices of an equilateral triangle. Then $a^2-3b =$

  1. $0$
  2. $1$
  3. $-1$
  4. $2$
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A Correct answer
Explanation

If roots z1, z2 and origin form an equilateral triangle, then z1^2 + z2^2 + 0^2 = z1*z2 + z1*0 + z2*0. So z1^2 + z2^2 = z1*z2. (z1+z2)^2 - 2z1z2 = z1z2 => (z1+z2)^2 = 3z1z2. From Vieta's, z1+z2 = -a and z1z2 = b. So (-a)^2 = 3b => a^2 = 3b => a^2 - 3b = 0.

AI explanation

By Vieta's formulas, the sum of the roots z1 + z2 equals -a, and the product of the roots z1*z2 equals b. The distance from the origin to the vertices z1 and z2 is their respective modulus, and because they form an equilateral triangle with the origin, |z1| = |z2|, which means the origin angle is 60 degrees. Using the distance formula for the third side, |z1 - z2|^2 = |z1|^2 + |z2|^2 - 2|z1||z2|cos(60) = |z1|^2, establishing that |z1 - z2|^2 = z1*z2. Expanding |z1 - z2|^2 as (z1 + z2)^2 - 4z1z2 and substituting the Vieta relations gives a^2 - 4b = b. Rearranging this equation gives the final result a^2 = 3b, meaning a^2 - 3b = 0.

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