(3-2i)(5+4i)-(3-4i)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (3-2i)(5+4i)-(3-4i)^2

This article will guide you through simplifying the complex expression: (3-2i)(5+4i)-(3-4i)^2. We will use the properties of complex numbers and algebraic operations to reach a solution.

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in 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 break down the simplification step by step:

1. Expand the first product: (3-2i)(5+4i) = 15 + 12i - 10i - 8i² Remember that i² = -1. Substitute this: = 15 + 12i - 10i + 8 = 23 + 2i

2. Expand the second product: (3-4i)² = (3-4i)(3-4i) = 9 - 12i - 12i + 16i² Substitute i² = -1: = 9 - 12i - 12i - 16 = -7 - 24i

3. Combine the results: (3-2i)(5+4i)-(3-4i)² = (23 + 2i) - (-7 - 24i)

4. Simplify: = 23 + 2i + 7 + 24i = 30 + 26i

Final Solution

Therefore, the simplified form of the expression (3-2i)(5+4i)-(3-4i)^2 is 30 + 26i.