Factoring Quadratic Expressions with Multiple Variables: A Comprehensive Guide

When dealing with quadratic expressions in multiple variables, the process of factorization can sometimes be complex and require careful attention to detail. In this article, we will explore the step-by-step process of factoring the quadratic expression x^2 3a4b x 2a^2 5ab 3b^2. This guide will cover the detailed methodology, key concepts, and practical examples to help you achieve a thorough understanding of the subject.

Understanding the Expression

The given quadratic expression is:

x2 3a4b x 2a25ab3b2

This expression can be more clearly written as:

x^2 - 3a 4b - x - 2a^2 - 5ab - 3b^2

Exploring the Process of Factorization

To factor this quadratic expression, we need to rearrange and group similar terms together. However, it's important to observe that the terms are not fully compatible for direct grouping due to the presence of different variables and coefficients.

Step 1: Rearrange and Group Terms

We can group the expression into parts that can be more easily factored:

x^2 - x - 3a - 2a^2 - 5ab - 3b^2

Step 2: Identify the Quadratic Form

Let's denote the quadratic in standard form:

x^2 Bx C

where:

B -1 - 3a - 5b C -3a - 2a^2 - 3b^2 - 5ab

Step 3: Check for Factorization

To check if we can factor, we need to find two numbers p and q such that:

p q B p * q C

Let's denote:

p q -1 - 3a - 5b p * q -3a - 2a^2 - 3b^2 - 5ab

Step 4: Factor the Quadratic Expression

Given the complexity of the expression, let's simplify it further by focusing on the constant term:

2a^2 - 5ab - 3b^2

The expression can be factored using the quadratic formula or by inspection. By inspection, we find that:

2a^2 - 5ab - 3b^2 (2a b)(a - 3b)

Step 5: Final Factorization

By using the factorization found in step 4, we can rewrite the original expression as:

x^2 - (2a b)x (2a b)(a - 3b)

Thus, the factorization of the expression is:

boxed{(x (2a b))(x - (a - 3b))}

Conclusion

The process of factoring quadratic expressions with multiple variables involves several steps, including rearranging the expression, identifying the quadratic form, checking for factorability, and applying factorization techniques. Understanding these steps can help simplify the process and achieve accurate results.

Key Takeaways

Regroup terms to simplify the expression. Identify the quadratic form. Check for factorability. Apply factorization techniques such as grouping or inspection.

For more information on factoring quadratic expressions, visit our resources page or explore additional examples in the practice section.