If ` z_(1) = 2+ 3i and z_(2) = 5-3i " then " z_(1)z_(2)` is

Options

A. `-9-19i`
B. `-9+19i`
C. ` 19-19`
D. `19 +i9`

Answer

To find the product \( z_1 z_2 \) where \( z_1 = 2 + 3i \) and \( z_2 = 5 - 3i \), we will use the distributive property (also known as the FOIL method for binomials).

### Step-by-Step Solution:

1. **Write down the complex numbers**:
\[ z_1 = 2 + 3i \]
\[ z_2 = 5 - 3i \]

2. **Multiply the two complex numbers**:
\[ z_1 z_2 = (2 + 3i)(5 - 3i) \]

3. **Apply the distributive property**:
\[ z_1 z_2 = 2 \cdot 5 + 2 \cdot (-3i) + 3i \cdot 5 + 3i \cdot (-3i) \]

4. **Calculate each term**:
- First term: \( 2 \cdot 5 = 10 \)
- Second term: \( 2 \cdot (-3i) = -6i \)
- Third term: \( 3i \cdot 5 = 15i \)
- Fourth term: \( 3i \cdot (-3i) = -9i^2 \)

5. **Combine the results**:
\[ z_1 z_2 = 10 - 6i + 15i - 9i^2 \]

6. **Recall that \( i^2 = -1 \)**:
\[ -9i^2 = -9(-1) = 9 \]

7. **Combine like terms**:
\[ z_1 z_2 = 10 + 9 + (-6i + 15i) \]
\[ z_1 z_2 = 19 + 9i \]

### Final Answer:
\[ z_1 z_2 = 19 + 9i \]