(5+2i)^2 - 10(5+2i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (5+2i)^2 - 10(5+2i)

This article will demonstrate the simplification of the complex expression (5+2i)^2 - 10(5+2i). We will walk through the steps involved in arriving at the solution using the fundamental properties of complex numbers.

Understanding Complex Numbers

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as √-1.

Simplifying the Expression

1. Expanding the Square:

   • (5+2i)^2 = (5+2i)(5+2i)
   • Using the FOIL (First, Outer, Inner, Last) method for expanding binomials, we get:
     • (5 * 5) + (5 * 2i) + (2i * 5) + (2i * 2i)
     • = 25 + 10i + 10i + 4i^2
   • Since i^2 = -1:
     • = 25 + 20i - 4
     • = 21 + 20i

2. Distributing the Constant:

   • -10(5+2i) = -50 - 20i

3. Combining Terms:

   • (21 + 20i) + (-50 - 20i) = 21 - 50 + 20i - 20i
   • = -29

Solution

Therefore, the simplified form of the expression (5+2i)^2 - 10(5+2i) is -29.

This example highlights the process of simplifying complex expressions, requiring the understanding of basic operations like multiplication and distribution, combined with the fundamental property of i^2 = -1.