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Multiplying Complex Conjugates: (x + 4i)(x - 4i)
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.
Understanding Complex Conjugates
- Complex Number: A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).
- Complex Conjugate: The complex conjugate of a + bi is a - bi.
Multiplying (x + 4i)(x - 4i)
We can multiply this expression using the FOIL method:
- First: x * x = x²
- Outer: x * -4i = -4xi
- Inner: 4i * x = 4xi
- Last: 4i * -4i = -16i²
Combining the terms:
x² - 4xi + 4xi - 16i²
Simplifying the Expression
- Recall: i² = -1
- Substitute: -16i² = -16 * (-1) = 16
Therefore, the simplified expression becomes:
x² + 16
Important Observation
The result of multiplying complex conjugates is always a real number. This is a useful property in many areas of mathematics and physics.
Key Takeaways
- Multiplying complex conjugates eliminates the imaginary term, resulting in a real number.
- Understanding complex conjugates is essential in simplifying complex expressions and solving various mathematical problems.