Problem 13 - Algebra 2

Question

Simplify each sum. \(\frac{5 x}{x^{2}-9}+\frac{2}{x+4}\)

Step-by-Step Solution

Answer

The simplified form of the given sum is \(\frac{7x - 6}{(x+3)(x-3)}\)

Step 1: Factor the Denominator

The first term in the expression has a denominator with a difference of squares. This can be factored using the identity \(a^{2}-b^{2} = (a+b)(a-b)\). Here, \(a = x\) and \(b = 3\). So \(x^{2}-9 = (x+3)(x-3)\). Thus the expression becomes \[\frac{5x}{(x+3)(x-3)} + \frac{2}{x+3}\]

Step 2: Find a Common Denominator

The common denominator of the two fractions is \((x+3)(x-3)\). Multiply the second term by \(\frac{(x-3)}{(x-3)}\) to give it the same denominator as the first term. This yields \[\frac{5x}{(x+3)(x-3)} + \frac{2(x-3)}{(x+3)(x-3)}\]

Step 3: Simplify the Numerator

Now that we have a common denominator, simplify the expression in the numerator by adding the like terms of the two fractions. \[\frac{5x + 2(x-3)}{(x+3)(x-3)}\] = \[\frac{5x + 2x - 6}{(x+3)(x-3)}\] = \[\frac{7x - 6}{(x+3)(x-3)}\]

Key Concepts

Factoring Expressions

Factoring is a key concept in algebra that helps simplify complex expressions. In this exercise, the first step involves factoring the difference of squares in the denominator \(x^2 - 9\). The difference of squares is a useful formula: \(a^2 - b^2 = (a+b)(a-b)\). It allows us to break down polynomials into simpler, more manageable terms. Here:

• Identify \(a\) and \(b\). In our example, \(a = x \) and \(b = 3\).
• Rewrite \(x^2 - 9\) as \((x+3)(x-3)\), using the formula.

Factoring not only makes the expression easier to work with but also assists in finding a common denominator, which is crucial for the next steps in solving fractions.

Common Denominator

To add or subtract algebraic fractions, having a common denominator is essential. The common denominator is the least common multiple of the denominators of the fractions involved. In our exercise, we need to make the fractions' denominators the same:

• The first fraction already has the denominator \((x+3)(x-3)\).
• The second fraction \(\frac{2}{x+4}\) needs to be adjusted to have the same denominator. Multiply both its numerator and denominator by \((x-3)\).

This results in both fractions having \((x+3)(x-3)\) as a common denominator, allowing the numerators to be combined in subsequent steps. A common denominator is essential, as it allows us to add or subtract fractions directly.

Simplifying Expressions

Once the fractions have a common denominator, the next challenge is simplifying the expression. Here, we combine the numerators of the fractions over the shared denominator. This involves:

• Adding the numerators: \(5x + 2(x-3)\).
• Distributing and combining like terms: \(5x + 2x - 6 = 7x - 6\).

The resulting expression \(\frac{7x - 6}{(x+3)(x-3)}\) is the simplified form of the given algebraic fraction. Simplifying expressions reduces complexity, offering a clearer and more concise result, and it is the final goal in problems involving algebraic fractions.