(x-4-5i)(x-4+5i)

3 min read Jun 17, 2024

Expanding Complex Number Expressions: (x - 4 - 5i)(x - 4 + 5i)

This article explores the process of expanding the complex number expression (x - 4 - 5i)(x - 4 + 5i). We'll utilize the distributive property and properties of complex numbers to simplify the expression.

Understanding the Problem

We're given a product of two binomials with complex coefficients. Our goal is to expand this expression and arrive at a simplified form, ideally eliminating any imaginary components.

Expanding the Expression

We can expand this expression using the distributive property (also known as FOIL - First, Outer, Inner, Last). Let's break it down step-by-step:

First: (x * x) = x²
Outer: (x * -4) = -4x
Inner: (-4 * x) = -4x
Last: (-4 * -4) = 16
First: (x * 5i) = 5ix
Outer: (-4 * 5i) = -20i
Inner: (-5i * x) = -5ix
Last: (-5i * 5i) = -25i²

Now, we have:

x² - 4x - 4x + 16 + 5ix - 20i - 5ix - 25i²

Simplifying the Expression

We can simplify this expression by combining like terms and remembering that i² = -1.

Combining real terms: x² - 8x + 16
Combining imaginary terms: 5ix - 5ix = 0
Simplifying -25i²: -25 * (-1) = 25

This leaves us with:

x² - 8x + 16 + 25

Finally, combining the constant terms gives us the simplified expression:

x² - 8x + 41

Conclusion

By expanding and simplifying the complex expression (x - 4 - 5i)(x - 4 + 5i), we arrive at the real quadratic expression x² - 8x + 41. This demonstrates how multiplying complex numbers can result in real-valued expressions.