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Expanding and Simplifying Complex Expressions: (x-7-5i)(x-7+5i)
This article explores the process of expanding and simplifying the complex expression (x-7-5i)(x-7+5i). We'll use the principles of complex number arithmetic and algebraic manipulation to achieve the simplified result.
Understanding Complex Numbers
A complex number is a number 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 (i² = -1).
Expanding the Expression
To expand the expression, we'll utilize the distributive property (also known as FOIL method) for multiplying binomials:
(x - 7 - 5i)(x - 7 + 5i) =
- x(x - 7 + 5i) - 7(x - 7 + 5i) - 5i(x - 7 + 5i)
Expanding further, we get:
- x² - 7x + 5ix - 7x + 49 - 35i - 5ix + 35i - 25i²
Simplifying the Expression
Notice that the terms 5ix and -5ix cancel each other out. Additionally, remember that i² = -1. Substituting this, we simplify the expression:
- x² - 14x + 49 - 25(-1)
Combining like terms, we get:
- x² - 14x + 49 + 25
Finally, the simplified expression is:
x² - 14x + 74
Conclusion
Expanding and simplifying the complex expression (x-7-5i)(x-7+5i) results in the real polynomial x² - 14x + 74. This demonstrates the power of complex number arithmetic in simplifying seemingly complex expressions.