Problem 31 - Elementary Algebra

Question

For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{3}{x-5}-\frac{4}{x^{2}-25}+\frac{5}{x+5} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{8x - 14}{(x-5)(x+5)}\).

Step 1: Identify the common denominator

The expressions in the problem all involve denominators related to the terms \(x-5\), \(x^2-25\), and \(x+5\). First, factor the quadratic denominator: \[ x^2 - 25 = (x - 5)(x + 5) \]Thus, the common denominator for all three fractions is \((x-5)(x+5)\).

Step 2: Adjust fractions to have the common denominator

For the first fraction, \( \frac{3}{x-5} \), we multiply the numerator and denominator by \(x+5\) to get:\[ \frac{3(x+5)}{(x-5)(x+5)} \]For the second fraction, \( \frac{4}{x^2-25} \), it's already in the form \( \frac{4}{(x-5)(x+5)} \).For the third fraction, \( \frac{5}{x+5} \), multiply the numerator and denominator by \(x-5\) to get:\[ \frac{5(x-5)}{(x-5)(x+5)} \]

Step 3: Combine the fractions

Now, combine all three fractions into a single fraction:\[ \frac{3(x+5) - 4 + 5(x-5)}{(x-5)(x+5)} \]

Step 4: Simplify the numerator

Expand and simplify the numerator: \[ 3(x+5) = 3x + 15 \]\[ 5(x-5) = 5x - 25 \]Combine the terms:\[ 3x + 15 - 4 + 5x - 25 \]Simplify further:\[ (3x + 5x) + (15 - 4 - 25) = 8x - 14 \]

Step 5: Write the final expression

Combine the simplified numerator back with the common denominator:\[ \frac{8x - 14}{(x-5)(x+5)} \]The expression cannot be simplified further since \(8x - 14\) and \((x-5)(x+5)\) do not have common factors.

Key Concepts

Common Denominator

When working with fractions in algebra, finding a common denominator is crucial. It allows you to combine fractions by giving them a shared base, simplifying the arithmetic.
First, you need to identify each denominator within the fractions. In our problem, they are related to the expressions \( x-5 \), \( x^2-25 \), and \( x+5 \).

• Recognize that \( x^2-25 \) is a difference of squares, which can be factored as \((x-5)(x+5)\).
• The common denominator is formed by multiplying all distinct factors of each denominator present.

In this case, the common denominator becomes \((x-5)(x+5)\). This is the baseline that lets us rewrite each fraction in a compatible form for addition or subtraction.

Simplifying Fractions

Once your fractions share a common denominator, you can proceed to simplify them. Simplification might involve rewriting fractions to expose common terms or removing any terms that are repeatedly present.
To simplify, you follow these steps:

• Ensure all fractions have the same denominator.
• Adjust each fraction by multiplying both numerator and denominator by the necessary factors to achieve the common denominator.
• Combine the numerators while retaining the common denominator.
• Simplify the resulting numerator by combining like terms.

For our problem, after factoring and adjusting, we reach the combined expression:
\[\frac{3(x+5) - 4 + 5(x-5)}{(x-5)(x+5)}\]
This setup is what allows you to merge the fractions seamlessly, leading to further simplification by expanding the terms.

Factoring Quadratics

Factoring quadratic expressions is a fundamental skill in algebra that facilitates simplification. The expression \(x^2 - 25\) in our problem is a classic example of a difference of squares, which can be easily factored.
Here's how you proceed with factoring quadratics like these:

• Identify the form: A difference of squares looks like \(a^2 - b^2\).
• Use the identity \(a^2 - b^2 = (a-b)(a+b)\).
• Apply this strategy: In our equation, \(x^2 - 25\) breaks down to \((x-5)(x+5)\).

Factoring reveals common denominators and simplifies expressions for both addition and subtraction tasks. It’s a key reason why algebraic manipulations remain manageable. Knowing how to quickly recognize and handle quadratic differences significantly eases working with algebraic fractions.