If `8 iz ^(3) + 12 z ^(2) - 18 z + 27i = 0,` then the vale of |z| is

Options

A. `3/2`
B. `2/3`
C. `1`
D. `3/4`

Answer

To solve the equation \( 8i z^3 + 12z^2 - 18z + 27i = 0 \) and find the value of \( |z| \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 8i z^3 + 12z^2 - 18z + 27i = 0 \]

### Step 2: Grouping Terms
We can group the terms to factor them:
\[ 8i z^3 + 12z^2 - 18z + 27i = 0 \]
Let's factor out \( 4z^2 \) from the first two terms and \( 9i \) from the last two terms:
\[ 4z^2(2iz + 3) + 9i(3 - 2iz) = 0 \]

### Step 3: Factoring the Equation
Now we can rewrite the equation:
\[ 4z^2(2iz + 3) + 9i(3 - 2iz) = 0 \]
This can be rearranged to:
\[ 4z^2(2iz + 3) + 9i(3 - 2iz) = 0 \]
We can set each factor to zero:
1. \( 4z^2 = 0 \)
2. \( 2iz + 3 = 0 \)

### Step 4: Solving the First Factor
From \( 4z^2 = 0 \):
\[ z^2 = 0 \implies z = 0 \]
However, we need to find \( |z| \), and \( |0| = 0 \) is not a valid solution in this context.

### Step 5: Solving the Second Factor
From \( 2iz + 3 = 0 \):
\[ 2iz = -3 \implies z = -\frac{3}{2i} \]
To simplify this, we can multiply the numerator and denominator by \( i \):
\[ z = -\frac{3i}{2i^2} = -\frac{3i}{-2} = \frac{3i}{2} \]

### Step 6: Finding the Modulus
Now we need to find \( |z| \):
\[ |z| = \left| \frac{3i}{2} \right| = \frac{3}{2} \]

### Conclusion
Thus, the value of \( |z| \) is:
\[ \boxed{\frac{3}{2}} \]