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Expanding the Complex Expression: (x-2-3i)(x-2+3i)
This expression involves multiplying two complex binomials. Let's break down the process and understand the result.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
Expanding the Expression
To multiply the given binomials, we can use the distributive property (also known as FOIL method):
- First: (x)(x) = x²
- Outer: (x)(3i) = 3xi
- Inner: (-2)(x) = -2x
- Last: (-2)(3i) = -6i
- Middle terms: (3i)(x) = 3xi and (-3i)(-2) = 6i
Combining these terms, we get:
x² + 3xi - 2x - 6i + 3xi + 6i
Simplifying the Expression
Notice that the terms with 'i' cancel out (3xi - 6i + 3xi + 6i = 0). This leaves us with:
x² - 2x
The Result
The product of (x-2-3i) and (x-2+3i) simplifies to x² - 2x. This result demonstrates that multiplying complex conjugates (numbers that differ only in the sign of their imaginary part) leads to a real number.