Complex Numbers
Solving Equations
Premium Question
Grade 11
Question:
The solutions of $z^2 + z|z| + |z|^2 = 0$ in $\mathbb{C}$ are:
(1) $z = 0$ only
(2) $z = 0$ or $\arg(z) = \pm\pi/3$
(3) $z = 0$ or $\arg(z) = \pm2\pi/3$
(4) $z$ purely imaginary
Step-by-Step Solution
Key Concept: Let $|z| = r$. Divide by $r^2$ (for $r \neq 0$): set $w = z/r$ so $|w| = 1$ and $w^2 + w + 1 = 0$. Solve for $w$ and find the argument.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)