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Understanding Complex Number Multiplication: (x-3i)(x+3i)
This article explores the multiplication of complex numbers, specifically focusing on the expression (x-3i)(x+3i). We'll delve into the process and the resulting outcome.
Complex Numbers: A Quick Review
Complex numbers involve the imaginary unit "i", where i² = -1. They are expressed in the form a + bi, where 'a' and 'b' are real numbers.
Multiplying Complex Numbers
Multiplying complex numbers follows the distributive property, similar to multiplying binomials with real numbers. We multiply each term in the first complex number by each term in the second complex number.
Applying the distributive property to (x-3i)(x+3i):
- x * x = x²
- x * 3i = 3xi
- -3i * x = -3xi
- -3i * 3i = -9i²
Combining these terms, we get: x² + 3xi - 3xi - 9i²
Simplifying the Expression
Notice that the terms 3xi and -3xi cancel each other out. We are left with:
x² - 9i²
Recall that i² = -1. Substituting this in, we get:
x² - 9(-1)
Finally, simplifying further:
x² + 9
Conclusion
The multiplication of (x-3i)(x+3i) results in x² + 9. This demonstrates a crucial concept in complex numbers: multiplying a complex number by its conjugate (the same number with the opposite sign for the imaginary part) always produces a real number. This is because the imaginary terms cancel out, leaving only a real number.