(2+i)(3-5i)-(1-4i)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (2+i)(3-5i)-(1-4i)^2

This article will walk through the process of simplifying the complex expression (2+i)(3-5i)-(1-4i)^2. We will utilize the distributive property, the concept of imaginary numbers, and the order of operations to arrive at a simplified form.

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 the square root of -1 (i² = -1).

Simplifying the Expression

Let's break down the expression step-by-step:

1. Expanding the products:

• (2+i)(3-5i):
  • Using the distributive property, we multiply each term in the first parenthesis by each term in the second parenthesis:
  • (2 * 3) + (2 * -5i) + (i * 3) + (i * -5i) = 6 - 10i + 3i - 5i²
• (1-4i)²:
  • Expanding the square: (1-4i)(1-4i)
  • Using the distributive property: (1 * 1) + (1 * -4i) + (-4i * 1) + (-4i * -4i) = 1 - 4i - 4i + 16i²

2. Simplifying using i² = -1:

• In both expanded expressions, we replace i² with -1:
  • 6 - 10i + 3i - 5(-1) = 6 - 10i + 3i + 5
  • 1 - 4i - 4i + 16(-1) = 1 - 4i - 4i - 16

3. Combining like terms:

• (2+i)(3-5i): 6 + 5 - 10i + 3i = 11 - 7i
• (1-4i)²: 1 - 16 - 4i - 4i = -15 - 8i

4. Final subtraction:

• (11 - 7i) - (-15 - 8i) = 11 - 7i + 15 + 8i = 26 + i

Result

Therefore, the simplified form of the expression (2+i)(3-5i)-(1-4i)² is 26 + i.