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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.