(x-3-i).jpeg "(x-3+i)(x-3-i)")
Multiplying Complex Conjugates: (x - 3 + i)(x - 3 - i)
This problem 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.
Here's how to multiply them:
Step 1: Recognize the Pattern
Notice that the two factors are complex conjugates:
- (x - 3 + i) has a real part of (x - 3) and an imaginary part of +1
- (x - 3 - i) has a real part of (x - 3) and an imaginary part of -1
Step 2: Apply the Difference of Squares
When multiplying complex conjugates, we can use the difference of squares pattern:
(a + b)(a - b) = a² - b²
In this case:
- a = (x - 3)
- b = i
Step 3: Expand
Applying the difference of squares pattern:
(x - 3 + i)(x - 3 - i) = (x - 3)² - i²
Step 4: Simplify
- Expand (x - 3)²: (x - 3)² = x² - 6x + 9
- Simplify i²: i² = -1
Therefore:
(x - 3)² - i² = x² - 6x + 9 - (-1)
Final Answer
(x - 3 + i)(x - 3 - i) = x² - 6x + 10
Key Points
- Multiplying complex conjugates always results in a real number.
- The difference of squares pattern is a useful shortcut for multiplying complex conjugates.
- The imaginary terms cancel out, leaving only the real terms in the final answer.