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Multiplying Complex Numbers: (x - 5 - 5i)(x - 5 + 5i)
This problem involves multiplying two complex numbers in the form of (a + bi) and (a - bi), which are known as complex conjugates. Complex conjugates have a special property that simplifies their multiplication: the product of two complex conjugates results in a real number.
Let's break down the multiplication step by step:
1. Expanding the Product
We can use the FOIL method (First, Outer, Inner, Last) to expand the product:
(x - 5 - 5i)(x - 5 + 5i) =
- First: x * x = x²
- Outer: x * (5i) = 5ix
- Inner: (-5 - 5i) * x = -5x - 5ix
- Last: (-5 - 5i) * (5i) = -25 - 25i²
2. Simplifying the Expression
Now, let's simplify the expression by combining like terms and remembering that i² = -1:
x² + 5ix - 5x - 5ix - 25 + 25 = x² - 5x + 0
3. Final Result
The final result of multiplying (x - 5 - 5i)(x - 5 + 5i) is:
x² - 5x
Key Takeaways:
- The product of two complex conjugates is always a real number.
- Multiplying complex numbers involves expanding the product and then simplifying it by combining like terms and using the fact that i² = -1.
This example demonstrates how the special property of complex conjugates simplifies the multiplication process and leads to a real number as the final result.