(x-3%2B3i).jpeg "(x-3-3i)(x-3+3i)")
Expanding and Simplifying (x - 3 - 3i)(x - 3 + 3i)
This expression involves complex numbers and can be simplified by using the distributive property (FOIL method) and recognizing that the product of a complex number and its conjugate results in a real number.
Here's how to expand and simplify:
1. Apply the FOIL method:
- First: (x)(x) = x²
- Outer: (x)(-3 + 3i) = -3x + 3xi
- Inner: (-3 - 3i)(x) = -3x - 3xi
- Last: (-3 - 3i)(-3 + 3i) = 9 - 9i²
2. Combine the terms and simplify:
x² - 3x + 3xi - 3x - 3xi + 9 - 9i²
3. Remember that i² = -1:
x² - 6x + 9 - 9(-1)
4. Simplify further:
x² - 6x + 9 + 9
5. Final result:
x² - 6x + 18
Key Takeaway
The product of a complex number and its conjugate always results in a real number. This is a useful property to remember when simplifying expressions involving complex numbers.