## Multiplying Complex Conjugates: A Simple Example

This article will explore the multiplication of the complex numbers **(x - 3 - 2i)** and **(x - 3 + 2i)**. We will demonstrate why this seemingly complex operation actually leads to a very simple result.

### Understanding Complex Conjugates

Before we dive into the multiplication, let's understand what makes these two numbers special. The numbers **(x - 3 - 2i)** and **(x - 3 + 2i)** are complex conjugates. This means they have the same real part (x - 3) but opposite imaginary parts (-2i and +2i).

### The Multiplication

Let's perform the multiplication:

**(x - 3 - 2i)(x - 3 + 2i)**

We can expand this using the distributive property (or FOIL method):

**x*** (x - 3 + 2i) = x² - 3x + 2xi**-3*** (x - 3 + 2i) = -3x + 9 - 6i**-2i*** (x - 3 + 2i) = -2xi + 6i - 4i²

Now, let's combine like terms. Notice that the terms with 'i' cancel each other out:

x² - 3x + 2xi - 3x + 9 - 6i - 2xi + 6i - 4i²

This simplifies to:

x² - 6x + 9 - 4i²

### Simplifying with i² = -1

Remember, the imaginary unit 'i' is defined as the square root of -1. Therefore, i² = -1. Substituting this into our expression:

x² - 6x + 9 - 4(-1)

This gives us:

**x² - 6x + 13**

### The Result

The multiplication of the complex conjugates **(x - 3 - 2i)** and **(x - 3 + 2i)** results in the real quadratic expression **x² - 6x + 13**. This outcome highlights a crucial property of complex conjugates: their product is always a real number.