(x-3%2B4i).jpeg "(x-3-4i)(x-3+4i)")
Multiplying Complex Conjugates: (x - 3 - 4i)(x - 3 + 4i)
This expression involves multiplying two complex numbers that are conjugates of each other. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts.
Let's break down the multiplication:
Understanding Complex Conjugates
- The conjugate of a complex number (a + bi) is (a - bi).
- The product of a complex number and its conjugate always results in a real number. This is a useful property in various mathematical contexts.
The Multiplication
-
FOIL Method: We can use the FOIL method (First, Outer, Inner, Last) to multiply the two binomials.
- First: (x)(x) = x²
- Outer: (x)(+4i) = +4ix
- Inner: (-3)(-4i) = +12i
- Last: (-3)(+4i) = -12i
-
Combining Terms: Now we have: x² + 4ix + 12i - 12i
-
Simplifying: Notice that the imaginary terms (+4ix and -12i) cancel out. This leaves us with: x² + 12
Conclusion
The product of (x - 3 - 4i) and (x - 3 + 4i) is x² + 12. As expected, the result is a real number. This illustrates the key property of complex conjugates – their product eliminates the imaginary terms, yielding a real number.