(x-1-2i)(x-1+2i)

Jasonbradley

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Jun 17, 2024

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Multiplying Complex Conjugates: (x-1-2i)(x-1+2i)

This expression involves multiplying two complex numbers that are complex conjugates. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.

Let's break down the steps:

1. Identify the complex conjugates:

   • (x-1-2i) and (x-1+2i) are complex conjugates because they have the same real part (x-1) and opposite imaginary parts (-2i and +2i).

2. Apply the difference of squares pattern:

   • The product of complex conjugates follows the difference of squares pattern: (a + b)(a - b) = a² - b²
   • In this case, a = (x - 1) and b = 2i

3. Substitute and simplify:

   • (x-1-2i)(x-1+2i) = (x-1)² - (2i)²
   • Expand: = x² - 2x + 1 - 4i²

4. Remember that i² = -1:

   • Substitute i² with -1: = x² - 2x + 1 - 4(-1)

5. Simplify further:

   • = x² - 2x + 1 + 4
   • = x² - 2x + 5

Therefore, the product of (x-1-2i) and (x-1+2i) is x² - 2x + 5.

Important Note: Multiplying complex conjugates always results in a real number. This is a useful property in simplifying complex expressions and solving equations.