(3-2i)(5+4i)-7i+1

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: A Step-by-Step Guide

This article will guide you through the process of simplifying the complex expression: (3-2i)(5+4i)-7i+1. We'll break down the steps, explaining the concepts involved.

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

1. Expanding the product:

   • Begin by multiplying the two complex numbers in the parentheses:

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

   • Apply the distributive property:

     = 15 + 12i - 10i - 8i²

2. Substituting i²:

   • Replace i² with -1:

     = 15 + 12i - 10i - 8(-1)

3. Combining real and imaginary terms:

   • Combine the real terms (15 and 8):

     = 23 + 12i - 10i

   • Combine the imaginary terms (12i and -10i):

     = 23 + 2i

4. Adding the remaining terms:

   • Add the remaining terms (-7i and 1) to the simplified result:

     = 23 + 2i - 7i + 1

   • Combine the real and imaginary terms:

     = 24 - 5i

Final Result

Therefore, the simplified form of the complex expression (3-2i)(5+4i)-7i+1 is 24 - 5i.