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

Expand the Square: (5 + 2i)² = (5 + 2i)(5 + 2i) = 25 + 10i + 10i + 4i²

Substitute i² with 1: 25 + 10i + 10i + 4i² = 25 + 10i + 10i + 4(1) = 21 + 20i

Distribute the 10: 10(5 + 2i) = 50  20i

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.