%5E2-10(5%2B2i).jpeg "(5+2i)^2-10(5+2i)")
Simplifying Complex Expressions: (5 + 2i)² - 10(5 + 2i)
This article will guide you through simplifying the complex expression (5 + 2i)² - 10(5 + 2i).
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 (i.e., i² = -1).
Steps to Simplify the Expression
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Expand the Square: (5 + 2i)² = (5 + 2i)(5 + 2i) = 25 + 10i + 10i + 4i²
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Substitute i² with -1: 25 + 10i + 10i + 4i² = 25 + 10i + 10i + 4(-1) = 21 + 20i
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Distribute the -10: -10(5 + 2i) = -50 - 20i
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Combine the terms: (21 + 20i) + (-50 - 20i) = -29
Final Result
The simplified form of the expression (5 + 2i)² - 10(5 + 2i) is -29.
Note: The result is a purely real number, meaning the imaginary component is zero.