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Simplifying Complex Expressions: (x-8-2i)(x-8+2i)
This expression involves multiplying two complex numbers. Let's break down the process of simplifying it:
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
The Key to Simplification: The Difference of Squares
The expression we are working with resembles the difference of squares pattern: (a + b)(a - b) = a² - b². We can apply this pattern here, considering a = (x - 8) and b = 2i.
Applying the Pattern
Let's perform the multiplication:
(x - 8 - 2i)(x - 8 + 2i) = [(x - 8) + (-2i)][(x - 8) + (2i)]
Following the difference of squares pattern, we get:
= (x - 8)² - (2i)²
Expanding and Simplifying
Now, we need to expand and simplify the expression:
= x² - 16x + 64 - (4 * i²)
Remember that i² = -1, so we substitute:
= x² - 16x + 64 - (4 * -1)
= x² - 16x + 68
Conclusion
The simplified form of the expression (x - 8 - 2i)(x - 8 + 2i) is x² - 16x + 68. This process showcases the power of recognizing patterns in mathematics and applying them to simplify complex expressions.