Factoring Quadratic Expressions using Algebraic Techniques

Quadratic expressions are a fundamental part of algebra, and understanding how to factor them correctly is crucial for solving complex problems. In this article, we will explore a specific quadratic expression a^2 - 4a32b - b^2 and demonstrate various techniques for factoring it.

Step-by-Step Analysis

Begin with the given expression:

a^2 - 4a32b - b^2

First, simplify the expression by breaking it down into manageable parts:

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

Combine like terms:

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

Further simplify by rearranging the terms:

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

Group the terms that can be factored:

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

Notice the structure of the expression and factor by grouping:

(a - b)(ab - 3 - b)

Using Simple Observation

Another method involves recognizing the structure as a standard form of a difference of squares:

a^2 - (4a32b b^2)

However, it is important to note that the expression contains a mistake in the original statement. It should be:

a^2 - 4ab - b^2

This expression can be factored as:

a^2 - 4ab - b^2 (a - 2b)^2 - b^2

Recognize it as a difference of squares:

((a - 2b) - b)((a - 2b) b) (a - 3b)(a - b)

Quadratic Expression in Terms of Variables

Consider the expression as a quadratic in terms of a:

a^2 - 4a - b^2 - 2b - 3

This can be written as:

a^2 - 4a - b^2 - 2b - 3 (a - 2b - 1)(a b - 3)

Find the roots of the quadratic equation:

The sum of the roots is 4. If one root is 3 - b, the other root must be:

4 - (3 - b) 1 b

Thus, the other factor is:

(a - (3 - b))(a - (1 b))

Polynomial Division

Another method involves polynomial division. Rewrite the expression as:

a^2 - 4a - b^2 - 2b - 3

Factor the quadratic expression:

a^2 - 4a - b^2 - 2b - 3 0

Solve for a by factorizing:

a 3 - b

Multiply the factors to check:

(a - (3 - b))(a - (1 b))

The other factor, after division, is:

a - 1 - b

Conclusion

Factoring quadratic expressions requires a combination of algebraic techniques, including grouping, recognizing patterns, and polynomial division. By understanding the underlying principles, one can efficiently solve complex problems and verify the factors of the given quadratic expression.

Note: Any mistakes or simplifications in the original expression will affect the final result, so always double-check the expression before proceeding with solving.