Find the sum and product of the complex number `(3-4i)` with its conjugate.

Options

A. `Sum =6, Product = 25 `
B. `Sum =25, Product = 5 `
C. `Sum =16, Product = 64`
D. none

Explanation

To find the sum and product of the complex number \( z = 3 - 4i \) with its conjugate, we will follow these steps:

### Step 1: Identify the complex number and its conjugate
The given complex number is:
\[ z = 3 - 4i \]
The conjugate of \( z \) is obtained by changing the sign of the imaginary part:
\[ \overline{z} = 3 + 4i \]

### Step 2: Calculate the sum of the complex number and its conjugate
To find the sum \( z + \overline{z} \):
\[ z + \overline{z} = (3 - 4i) + (3 + 4i) \]
Combine the real parts and the imaginary parts:
\[ = 3 + 3 + (-4i + 4i) = 6 + 0i = 6 \]

### Step 3: Calculate the product of the complex number and its conjugate
To find the product \( z \cdot \overline{z} \):
\[ z \cdot \overline{z} = (3 - 4i)(3 + 4i) \]
We can use the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
Here, \( a = 3 \) and \( b = 4i \):
\[ = 3^2 - (4i)^2 \]
Calculating each term:
\[ = 9 - 16(-1) = 9 + 16 = 25 \]

### Final Results
The sum of the complex number and its conjugate is:
\[ \text{Sum} = 6 \]
The product of the complex number and its conjugate is:
\[ \text{Product} = 25 \]