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Understanding Complex Number Multiplication: (x - 6i)(x + 6i)
This expression involves the multiplication of two complex numbers: (x - 6i) and (x + 6i). Let's break down the process and understand the result.
The Concept of Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers.
- i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
Expanding the Expression
To multiply the given complex numbers, we use the distributive property (also known as FOIL):
(x - 6i)(x + 6i) = x(x + 6i) - 6i(x + 6i)
Expanding further:
= x² + 6xi - 6xi - 36i²
Simplifying the Result
Notice that the terms 6xi and -6xi cancel each other out. Also, remember that i² = -1. Substituting this:
= x² - 36(-1)
= x² + 36
Conclusion
Therefore, the product of (x - 6i) and (x + 6i) is x² + 36. This result demonstrates a crucial concept in complex numbers: the product of a complex number and its conjugate (obtained by changing the sign of the imaginary part) always results in a real number.
This concept is particularly important in simplifying complex fractions and solving equations involving complex numbers.