Factoring Quadratic Expressions: A Comprehensive Guide

Understanding how to factor quadratic expressions is a fundamental skill in algebra, often critical for solving more complex equations. This guide will help you understand the process of factoring the expression 2a2 - 3a - 2b 2ab 1, along with various other techniques to simplify similar expressions.

Introduction to Factoring Quadratic Expressions

Quadratic expressions are polynomials of the second degree. They can be factored into simpler forms, which often helps in solving equations or simplifying calculations. The expression in our discussion, 2a2 - 3a - 2b 2ab 1, can be factored using different methods, including the quadratic formula and grouping techniques.

Step-by-Step Solution

Let's consider the given expression: 2a2 - 3a - 2b 2ab 1

Step 1: Rearrange and Group Terms

First, we rewrite the expression by rearranging and grouping the terms for easier factoring:

2a2   2ab - 3a - 2b   1

Step 2: Treating as a Quadratic in Terms of a

We can now treat the expression as a quadratic in (a):
2a2 2b - 3a - 2b 1

To use the quadratic formula, we identify the coefficients:

The quadratic formula is:
a -B ± √(B2 - 4AC)/

2A)

Calculate the Discriminant:

B2 - 4AC  (2b - 3)2 - 4(2)(-2b   1)

Expanding:

 4b2 - 12b   9   16b - 8

Resulting in:
4b2 4b 1

Therefore:
a (-(2b - 3) ± √(4b2 4b 1))/4

Which simplifies to:

Therefore, the factored form is:

2a2 - 3a - 2b   2ab   1  (2a - 1)(a - b - 1/2)

So, the factored form of the expression is 2a - 1a - b - 1/2.

Step 3: Verification

To verify the correctness of the factorization, we expand the product:

(2a - 1)(a - b - 1/2)  2a^2 - 2ab - a - 2a   b   1  2a^2 - 3a - 2b   2ab   1

This confirms that the factorization is correct.

Alternative Techniques

Instead of using the quadratic formula, we can use grouping techniques to factor the expression. Let's consider the expression: 2a^2 - 3a - 2b 2ab 1

Step 1: Rearrange:

2a^2 - 2a - a - 2b   1   2ab

Step 2: Group Terms:

2a^2   2ab - 2a - 2b - a - 1

Step 3: Factor Out Common Factors:

 a(2a   2b - 1) - (2a   2b   1)

Recognizing that the expression is not fully factorable using simple techniques, we revert to the quadratic formula approach for accuracy.

Conclusion

Factoring quadratic expressions involves systematic simplification and verification. By using the quadratic formula, we can systematically approach complex expressions like 2a2 - 3a - 2b 2ab 1. Always verify the factorization by expanding the product to ensure its correctness.

Keywords: factoring quadratic expressions, quadratic formula, factoring techniques