(-3i+4)(3i+4)

2 min read Jun 16, 2024

Multiplying Complex Numbers: A Walkthrough of (-3i + 4)(3i + 4)

This article explores how to multiply complex numbers, specifically focusing on the expression (-3i + 4)(3i + 4).

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).

Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property (often referred to as FOIL for first, outer, inner, last). This means we multiply each term in the first complex number by each term in the second complex number.

The Calculation

Let's break down the multiplication of (-3i + 4)(3i + 4):

First: (-3i) * (3i) = -9i²
Outer: (-3i) * (4) = -12i
Inner: (4) * (3i) = 12i
Last: (4) * (4) = 16

Now, we combine the terms: -9i² - 12i + 12i + 16

Since i² = -1, we can simplify: -9(-1) - 12i + 12i + 16

This simplifies further to: 9 + 16

Therefore, (-3i + 4)(3i + 4) = 25

Conclusion

We have successfully multiplied the complex numbers (-3i + 4) and (3i + 4) using the distributive property and the knowledge that i² = -1. The result, 25, is a real number. This demonstrates that multiplying complex numbers can sometimes lead to real number results.