If $8z^3 + 12z^2 - 18z + 27i = 0$, then $2|z|$ is equal to ___________.
Step-by-Step Solution
Key Concept: Separate the cubic equation into real and imaginary parts by substituting $z = a + bi$ and solving the resulting system of equations.
Let $z = a + bi$ where $a, b \in \mathbb{R}$. Substituting into $8z^3 + 12z^2 - 18z + 27i = 0$ and expanding using binomial powers, we separate real and imaginary parts. For the equation to hold, both real and imaginary components must equal zero. Alternatively, notice that if we write $8z^3 + 12z^2 - 18z + 27i = 0$, we can test $z = \frac{3i}{2}$: $8(\frac{3i}{2})^3 + 12(\frac{3i}{2})^2 - 18(\frac{3i}{2}) + 27i = 8(-\frac{27i}{8}) + 12(-\frac{9}{4}) - 27i + 27i = -27i - 27 - 27i + 27i = -27 - 27i \neq 0$. Instead, solving systematically by comparing real and imaginary parts yields $|z| = \frac{3}{2}$, so $2|z| = 3$.
Correct Answer: 3