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How many complex numbers z satisfy the equation z³ + (3|z|²)/(z) = 0, given that |z| = √(3)?

1. 2
2. 3
3. 6
4. 4

Correct answer: 4

Solution

With |z|=sqrt(3), |z|^2=3, so z^3 + 3*3/z = z^3 + 9/z = 0, giving z^4 = -9. This quartic has exactly 4 roots, each of modulus 9^(1/4)=sqrt(3), so 4 complex numbers satisfy it.

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