The roots of the equation `z^(4) + az^(3) + (12 + 9i)z^(2) + bz = 0` (where a and b are complex numbers) are the vertices of a square. Then The area of the square is

Options

A. 25 sq.units
B. 20 sq.units
C. 5 sq.unit
D. 4 sq .units

Explanation

To find the area of the square formed by the roots of the equation \( z^4 + az^3 + (12 + 9i)z^2 + bz = 0 \), we can follow these steps:

### Step 1: Factor out \( z \)
The given equation can be factored as:
\[ z(z^3 + az^2 + (12 + 9i)z + b) = 0 \]
This shows that one root is \( z = 0 \). The other three roots will be the roots of the cubic polynomial \( z^3 + az^2 + (12 + 9i)z + b = 0 \).

### Step 2: Identify the roots
Since the roots are the vertices of a square, we can denote the non-zero roots as \( z_1, z_2, z_3 \). For a square, if one vertex is at the origin (0), the other vertices can be represented as:
- \( z_1 = r \)
- \( z_2 = ri \)
- \( z_3 = r + ri \)

Here, \( r \) is the distance from the origin to the vertex \( z_1 \).

### Step 3: Use Vieta's formulas
According to Vieta's formulas for a cubic polynomial \( z^3 + az^2 + (12 + 9i)z + b = 0 \):
1. The sum of the roots \( z_1 + z_2 + z_3 = -a \)
2. The sum of the products of the roots taken two at a time \( z_1z_2 + z_2z_3 + z_3z_1 = 12 + 9i \)
3. The product of the roots \( z_1z_2z_3 = -b \)

### Step 4: Calculate the sum of the roots
Substituting the roots:
\[ z_1 + z_2 + z_3 = r + ri + (r + ri) = 2r + 2ri = 2r(1 + i) \]
Thus, we have:
\[ -a = 2r(1 + i) \implies a = -2r(1 + i) \]

### Step 5: Calculate the sum of the products of the roots
Calculating the sum of the products:
\[ z_1 z_2 = r(ri) = r^2 i \]
\[ z_2 z_3 = (ri)(r + ri) = r^2 i + r^2 i^2 = r^2 i - r^2 = r^2(i - 1) \]
\[ z_3 z_1 = (r + ri)r = r^2 + r^2 i \]
Adding these:
\[ z_1 z_2 + z_2 z_3 + z_3 z_1 = r^2 i + r^2(i - 1) + (r^2 + r^2 i) = 3r^2 i - r^2 \]
Setting this equal to \( 12 + 9i \):
\[ 3r^2 i - r^2 = 12 + 9i \]
This gives us two equations:
1. \( -r^2 = 12 \) (real part)
2. \( 3r^2 = 9 \) (imaginary part)

### Step 6: Solve for \( r^2 \)
From \( 3r^2 = 9 \):
\[ r^2 = 3 \]
From \( -r^2 = 12 \), we find a contradiction. Thus, we need to focus on the imaginary part:
\[ r^2 = 3 \]

### Step 7: Calculate the area of the square
The area \( A \) of the square is given by:
\[ A = \text{side}^2 \]
The side of the square is the distance from the origin to \( z_1 \):
\[ \text{side} = r = \sqrt{3} \]
Thus,
\[ A = (\sqrt{3})^2 = 3 \]

### Final Answer
The area of the square is \( \boxed{3} \).