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Multiplying Complex Conjugates: (x - 4 - 3i)(x - 4 + 3i)
This expression represents the product of two complex numbers that are conjugates of each other. Complex conjugates have the same real part but opposite imaginary parts. Let's explore how multiplying these conjugates simplifies to a real number.
Understanding Complex Conjugates
The conjugate of a complex number a + bi is a - bi. In our example, the complex numbers are:
- (x - 4 - 3i)
- (x - 4 + 3i)
Notice how the imaginary parts differ only in their sign.
The Multiplication Process
We can use the distributive property (or FOIL method) to multiply these complex numbers:
(x - 4 - 3i)(x - 4 + 3i) =
x(x - 4 + 3i) - 4(x - 4 + 3i) - 3i(x - 4 + 3i)
Expanding each term:
= x² - 4x + 3xi - 4x + 16 - 12i - 3xi + 12i - 9i²
Simplifying by combining like terms and remembering that i² = -1:
= x² - 8x + 16 - 9(-1)
= x² - 8x + 16 + 9
= **x² - 8x + 25**
Key Takeaway
The product of complex conjugates always results in a real number. This is a useful property for simplifying expressions involving complex numbers.