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Multiplying Complex Conjugates: A Walkthrough
This article explores the multiplication of complex conjugates, focusing on the expression (x - 1 - 3i)(x - 1 + 3i).
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In our case, (x - 1 - 3i) and (x - 1 + 3i) are complex conjugates.
Multiplying the Expression
We can multiply the given expression using the distributive property or the FOIL method:
(x - 1 - 3i)(x - 1 + 3i) =
- x(x - 1 + 3i) - 1(x - 1 + 3i) - 3i(x - 1 + 3i)
Expanding this further:
- x² - x + 3xi - x + 1 - 3i - 3xi + 3i - 9i²
Notice that the terms with 'i' cancel each other out, leaving us with:
- x² - 2x + 1 - 9i²
Recall that i² = -1. Substituting this value:
- x² - 2x + 1 + 9
Combining like terms gives us the final result:
x² - 2x + 10
Significance of the Result
The product of complex conjugates always results in a real number. This is a crucial property in various mathematical contexts, particularly in simplifying complex expressions and solving equations.
In this example, the initial complex expression, when multiplied by its conjugate, yielded a simple quadratic expression with only real terms. This illustrates the usefulness of complex conjugates in simplifying complex expressions and revealing their real-valued components.