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

2 min read Jun 17, 2024
(x-3-2i)(x-3+2i)

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.

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