Problem 44 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{x}{x^{2}-4}-\frac{5}{x^{2}-4 x+4}\)

Step-by-Step Solution

Answer

Combine fractions with the common denominator: \(\frac{x^2 - 7x - 10}{(x+2)(x-2)^2}\).

Step 1: Analyze the Denominators

The denominators of the fractions are \(x^{2}-4\) and \(x^{2} - 4x + 4\). Factor these expressions to simplify: \(x^{2} - 4\) is \((x+2)(x-2)\) and \(x^{2} - 4x + 4\) is \((x-2)(x-2)\). So, the expressions become: \(\frac{x}{(x+2)(x-2)} - \frac{5}{(x-2)^2}\).

Step 2: Find a Common Denominator

The common denominator between \((x+2)(x-2)\) and \((x-2)^2\) is \((x+2)(x-2)^2\). Convert each fraction to have this common denominator.

Step 3: Adjust First Fraction

The first fraction \(\frac{x}{(x+2)(x-2)}\) needs to be multiplied by \((x-2)\) in both numerator and denominator to achieve the common denominator: \(\frac{x(x-2)}{(x+2)(x-2)^2}\).

Step 4: Adjust Second Fraction

The second fraction \(\frac{5}{(x-2)^2}\) needs to be multiplied by \((x+2)\) in both numerator and denominator to achieve the common denominator: \(\frac{5(x+2)}{(x+2)(x-2)^2}\).

Step 5: Subtract the Fractions

Now that both fractions have the same common denominator, subtract the numerators: \( \frac{x(x-2) - 5(x+2)}{(x+2)(x-2)^2} \).

Step 6: Simplify the Expression

Simplify the numerator: \(x(x-2) = x^2 - 2x\) and \(-5(x+2) = -5x - 10\). Combine: \(x^2 - 2x - 5x - 10 = x^2 - 7x - 10\). So, \(\frac{x^2 - 7x - 10}{(x+2)(x-2)^2}\).

Step 7: Final Expression

The expression \(\frac{x^2 - 7x - 10}{(x+2)(x-2)^2}\) cannot be simplified further since the numerator does not factor nicely over the integers.

Key Concepts

Factoring Quadratics

Factoring quadratics is essential when working with algebraic fractions, particularly when simplifying or finding common denominators.
Quadratics take the form of a polynomial equation where the highest power of the variable is 2, such as \(ax^2 + bx + c\).

• For the quadratic \(x^2 - 4\), you can recognize it as a difference of squares. It factors into \((x+2)(x-2)\).
• The quadratic \(x^2 - 4x + 4\) is a perfect square trinomial, which can be written as \((x-2)(x-2)\) or \((x-2)^2\).

Factoring helps to identify common factors in expressions, which is crucial for operations like addition, subtraction, or simplification of algebraic fractions. Understanding how to factor different forms is critical to solving complex algebra problems.

Common Denominator

One of the fundamental steps when dealing with algebraic fractions is finding a common denominator. This is necessary when you are adding or subtracting fractions. By finding a common denominator, you can rewrite the fractions so that they have the same base, making it possible to perform the desired arithmetic operation.
In our example:

• The denominators are \((x+2)(x-2)\) and \((x-2)^2\).
• The least common denominator (LCD) would be \((x+2)(x-2)^2\). This combines all unique factors from both denominators and ensures that each original fraction's denominator can be converted to this form.

Finding a common denominator is like leveling the playing field so you can effectively operate on fractions, regardless of their original denominator. It is a crucial skill in algebra that allows more advanced manipulations of expressions.

Subtracting Fractions

Subtracting fractions with algebraic expressions follows the same principles as numeric fractions. Once a common denominator is established, you can directly subtract the numerators.
In the problem, the fractions are \(\frac{x(x-2)}{(x+2)(x-2)^2}\) and \(\frac{5(x+2)}{(x+2)(x-2)^2}\). Both fractions now share the common denominator \((x+2)(x-2)^2\).

• Subtract the numerators: \(x(x-2) - 5(x+2)\).

This results in a new expression where the like terms (those in the numerators) can be combined. Understanding how to subtract fractions simplifies the process of finding solutions to polynomial equations and rational expressions.

Simplifying Expressions

Simplification is the process of making an algebraic expression easier to handle, often by reducing the number of terms or making the expression more compact.
Once you've subtracted the fractions in the algebra problem, the next step is to simplify the resulting expression.
For the given expression \(x(x-2) - 5(x+2)\),

• Distribute and simplify: \(x^2 - 2x\) and \(-5x - 10\).
• Combine like terms to achieve the simplified form: \(x^2 - 7x - 10\).

The final expression, \(\frac{x^2 - 7x - 10}{(x+2)(x-2)^2}\), is mostly simplified as far as possible. Simplicity in an expression aids in understanding and further arithmetic operations, ensuring clarity and accuracy when solving algebraic problems.