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

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Number Expressions: (5 + 2i)² + (2 - i)²

This article will guide you through the process of simplifying the expression (5 + 2i)² + (2 - i)². We'll break down the steps involved, using the properties of complex numbers.

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 √-1.

Simplifying the Expression

Let's simplify the expression step-by-step:

1. Expand the Squares:

   • (5 + 2i)² = (5 + 2i)(5 + 2i) = 25 + 10i + 10i + 4i²
   • (2 - i)² = (2 - i)(2 - i) = 4 - 2i - 2i + i²

2. Simplify using i² = -1:

   • (5 + 2i)² = 25 + 20i + 4(-1) = 21 + 20i
   • (2 - i)² = 4 - 4i + (-1) = 3 - 4i

3. Combine the results:

   • (5 + 2i)² + (2 - i)² = (21 + 20i) + (3 - 4i)

4. Combine real and imaginary terms:

   • (21 + 20i) + (3 - 4i) = (21 + 3) + (20 - 4)i

5. Final Result:

   • (5 + 2i)² + (2 - i)² = 24 + 16i

Therefore, the simplified form of the expression (5 + 2i)² + (2 - i)² is 24 + 16i.