(x-2%2B5i).jpeg "(x-2-5i)(x-2+5i)")
Multiplying Complex Conjugates: A Simple Example
This article will demonstrate how to multiply two complex conjugates: (x - 2 - 5i) and (x - 2 + 5i).
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In our example, the real part of both numbers is (x - 2), and the imaginary parts are -5i and +5i, respectively.
Multiplication
Let's multiply the two complex numbers:
(x - 2 - 5i)(x - 2 + 5i)
We can use the FOIL method (First, Outer, Inner, Last) to expand this expression:
- First: (x)(x) = x²
- Outer: (x)(5i) = 5xi
- Inner: (-2)(-5i) = 10i
- Last: (-5i)(5i) = -25i²
Now, let's combine the terms:
x² + 5xi + 10i - 25i²
Remember that i² = -1. Substituting this value:
x² + 5xi + 10i - 25(-1)
Simplifying further:
x² + 5xi + 10i + 25
Notice that the imaginary terms (5xi and 10i) cancel each other out, leaving us with a purely real expression:
x² + 25
Conclusion
Multiplying complex conjugates always results in a real number. This is because the imaginary terms cancel each other out due to their opposite signs. In this case, the product of (x - 2 - 5i) and (x - 2 + 5i) is simply x² + 25.