(x-3%2Bi).jpeg "(x-3-i)(x-3+i)")
Multiplying Complex Conjugates: (x - 3 - i)(x - 3 + i)
This expression involves multiplying two complex numbers that are conjugates of each other. This is a common operation in complex algebra and often leads to simplifying the expression. Let's break down the steps:
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In this case, our complex numbers are:
- (x - 3 - i)
- (x - 3 + i)
Notice that the only difference is the sign of the imaginary part.
The Multiplication Process
We can multiply these complex numbers using the distributive property (or FOIL method):
- Multiply the first terms: (x - 3) * (x - 3) = x² - 6x + 9
- Multiply the outer terms: (x - 3) * (+i) = +ix - 3i
- Multiply the inner terms: (-i) * (x - 3) = -ix + 3i
- Multiply the last terms: (-i) * (+i) = -i²
Now, let's combine the terms:
x² - 6x + 9 + ix - 3i - ix + 3i - i²
Notice that the terms with 'i' cancel out: +ix - ix and -3i + 3i.
We also know that i² = -1, so we can substitute:
x² - 6x + 9 - (-1)
The Simplified Result
Finally, simplifying the expression gives us:
x² - 6x + 10
This is the result of multiplying the complex conjugates (x - 3 - i) and (x - 3 + i). As you can see, the final result is a real quadratic expression.
Key takeaway:
Multiplying complex conjugates always eliminates the imaginary component, resulting in a real number or expression. This property is often used in simplifying expressions involving complex numbers.