Simplifying Complex Expressions: A Comprehensive Guide to Algebraic Fractions

This article aims to provide a step-by-step guide on simplifying complex algebraic expressions, specifically focusing on the case of the given expression (frac{x^2 - 3x - 4}{x^2 - 4x} - frac{x^2 - 4x - 4}{x^2 - 4}). We will break down the process of simplification into manageable steps, explaining each step clearly and providing detailed calculations.

Introduction to Algebraic Fractions

Algebraic fractions are fractions where the numerator and the denominator are polynomials. Simplifying such fractions involves a series of steps including factoring, canceling common terms, and combining expressions. This process is crucial in solving equations and simplifying complex expressions in algebra.

Simplifying the Given Expression

The given expression is (frac{x^2 - 3x - 4}{x^2 - 4x} - frac{x^2 - 4x - 4}{x^2 - 4}). Following the steps, we will break it down to a simpler form.

Step 1: Factor Each Part of the Expression

Numerator of the First Fraction:

The numerator is (x^2 - 3x - 4)

To factor this, we need two numbers that multiply to (-4) and add to (-3). These numbers are (-4) and (1).

Therefore, (x^2 - 3x - 4 (x - 4)(x 1))

Denominator of the First Fraction:

The denominator is (x^2 - 4x)

This can be factored as (x(x - 4))

Numerator of the Second Fraction:

The numerator is (x^2 - 4x - 4)

Denominator of the Second Fraction:

The denominator is (x^2 - 4)

This is a difference of squares, which factors as ((x - 2)(x 2))

Step 2: Rewrite the Expression with Factored Forms

Now, substituting the factored forms back into the original expression, we have:

[frac{(x - 4)(x 1)}{x(x - 4)} - frac{(x - 2)^2}{(x - 2)(x 2)}]

Step 3: Simplify Each Fraction

First Fraction:

Simplify the first fraction by canceling the common factor ((x - 4)).

[frac{(x - 4)(x 1)}{x(x - 4)} frac{x 1}{x} quad text{for } x eq 4] Second Fraction:

Simplify the second fraction by canceling the common factor ((x - 2)).

[frac{(x - 2)^2}{(x - 2)(x 2)} frac{x - 2}{x 2} quad text{for } x eq 2]

Step 4: Combine the Simplified Fractions

Next, we combine the simplified fractions. The common denominator for (x) and ((x 2)) is (x(x 2)).

[frac{x 1}{x} - frac{x - 2}{x 2} frac{(x 1)(x 2) - x(x - 2)}{x(x 2)}]

Step 5: Expand and Simplify the Numerator

Expanding the numerator, we get:

[(x 1)(x 2) x^2 3x 2] [x(x - 2) x^2 - 2x] [x^2 3x 2 - (x^2 - 2x) 5x 2]

Step 6: Final Expression

Putting it all together, the final simplified expression is:

[frac{5x 2}{x(x 2)}]

Conclusion

The process of simplifying the given complex algebraic expression involves several steps including factoring, canceling out common terms, and performing arithmetic operations on the numerators. The final simplified form is (frac{5x 2}{x(x 2)}).

This expression is valid for (x eq 0, 2, -2) to avoid division by zero.