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(-2+3i)(4-3i)+6(3-5i)

2 min read Jun 16, 2024
(-2+3i)(4-3i)+6(3-5i)

Simplifying Complex Expressions: A Step-by-Step Guide

This article will guide you through simplifying the complex expression (-2 + 3i)(4 - 3i) + 6(3 - 5i).

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 (i.e., i² = -1).

Simplifying the Expression

Let's break down the simplification process step-by-step:

  1. Expand the first product: (-2 + 3i)(4 - 3i) = (-2)(4) + (-2)(-3i) + (3i)(4) + (3i)(-3i) = -8 + 6i + 12i - 9i²

  2. Substitute i² with -1: -8 + 6i + 12i - 9i² = -8 + 6i + 12i - 9(-1) = -8 + 6i + 12i + 9

  3. Combine real and imaginary terms: -8 + 6i + 12i + 9 = ( -8 + 9 ) + ( 6 + 12)i = 1 + 18i

  4. Expand the second product: 6(3 - 5i) = 6(3) + 6(-5i) = 18 - 30i

  5. Add the simplified products: 1 + 18i + 18 - 30i = (1 + 18) + (18 - 30)i = 19 - 12i

The Final Result

Therefore, the simplified form of the complex expression (-2 + 3i)(4 - 3i) + 6(3 - 5i) is 19 - 12i.

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