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

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

Multiplying Complex Numbers: A Step-by-Step Guide

This article will walk you through the process of multiplying two complex numbers, specifically: (-2 + 3i)(3 + 5i)

Understanding Complex Numbers

A complex number is a number 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.

Multiplying Complex Numbers

To multiply complex numbers, we can use the distributive property (also known as FOIL) similar to multiplying binomials.

Let's break down the multiplication of (-2 + 3i)(3 + 5i):

  1. FOIL: Multiply each term in the first complex number with each term in the second complex number:

    • (-2 * 3) + (-2 * 5i) + (3i * 3) + (3i * 5i)

  2. Simplify:

    • -6 -10i + 9i + 15i²

  3. Remember i² = -1:

    • -6 -10i + 9i + 15(-1)

  4. Combine real and imaginary terms:

    • (-6 -15) + (-10i + 9i)

  5. Final Result:

    • -21 - i

Therefore, (-2 + 3i)(3 + 5i) = -21 - i.

Key Points to Remember:

  • FOIL: Use the distributive property (FOIL) to multiply each term of the first complex number with each term of the second.
  • i² = -1: Substitute i² with -1 when simplifying the expression.
  • Combine Real and Imaginary: Group the real terms and the imaginary terms together to express the final answer in the form a + bi.

By following these steps, you can confidently multiply any two complex numbers.

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