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Multiplying Complex Numbers: (x-2i)(x+2i)
This article explores the multiplication of the complex numbers (x-2i) and (x+2i). We'll demonstrate how to perform the multiplication and discuss the resulting expression's significance.
Understanding Complex Numbers
Before diving into the multiplication, let's clarify what complex numbers are. Complex numbers are numbers that extend the real number system by including the imaginary unit "i", where i² = -1. A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers.
Multiplying (x-2i)(x+2i)
We can multiply the complex numbers (x-2i) and (x+2i) using the distributive property (also known as FOIL - First, Outer, Inner, Last):
(x-2i)(x+2i) = x(x+2i) -2i(x+2i)
Expanding this further:
= x² + 2ix - 2ix - 4i²
Since i² = -1, we can substitute:
= x² + 2ix - 2ix + 4
Simplifying the expression:
= x² + 4
Significance of the Result
The product of (x-2i) and (x+2i) is a real number, x² + 4. This outcome reveals an important property of complex numbers:
When multiplying a complex number by its complex conjugate, the result is always a real number.
The complex conjugate of a complex number 'a + bi' is 'a - bi'. In our case, (x-2i) and (x+2i) are complex conjugates of each other. This property is frequently utilized in simplifying complex expressions and solving equations involving complex numbers.
Conclusion
We've seen how to multiply the complex numbers (x-2i) and (x+2i), resulting in the real number x² + 4. Understanding this multiplication and the concept of complex conjugates is crucial for working with complex numbers and their applications in various fields, including mathematics, physics, and engineering.