Problem 85 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{x}{x^{2}-9}+\frac{5 x}{x-3} $$

Step-by-Step Solution

Answer

\(\frac{x(5x + 16)}{(x-3)(x+3)}\)

Step 1: Find the Common Denominator

To simplify the expression \( \frac{x}{x^{2}-9} + \frac{5x}{x-3} \), we first need to find a common denominator. The first denominator \( x^2 - 9 \) can be factored as \((x-3)(x+3)\). The second denominator is already \(x-3\). The common denominator of both fractions is \((x-3)(x+3)\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction so they both have the common denominator \((x-3)(x+3)\). The first fraction is already \( \frac{x}{(x-3)(x+3)} \). For the second fraction, multiply both the numerator and the denominator by \((x+3)\): \( \frac{5x(x+3)}{(x-3)(x+3)} \).

Step 3: Combine the Fractions

With both fractions having the common denominator, you can combine them: \( \frac{x + 5x(x+3)}{(x-3)(x+3)} \).

Step 4: Simplify the Numerator

Simplify the numerator \(x + 5x(x+3)\). Distribute the \(5x\) in the term \(5x(x+3)\): \(5x^2 + 15x\). The expression becomes \(x + 5x^2 + 15x\) which simplifies to \(5x^2 + 16x\).

Step 5: Factor the Numerator

To factor the numerator \(5x^2 + 16x\), factor out \(x\): \(x(5x + 16)\).

Step 6: Final Expression

The expression is now \(\frac{x(5x + 16)}{(x-3)(x+3)}\). This is the simplified form of the given expression since there are no common factors to cancel out between the numerator and the denominator.

Key Concepts

Common Denominator

When simplifying fractions, finding a common denominator is crucial. It ensures both fractions can be combined seamlessly. In algebra, variables and expressions form our denominators. Let's use the expression \( \frac{x}{x^{2}-9} + \frac{5x}{x-3} \).

We first determine a denominator that both fractions share. The denominator \( x^2 - 9 \) is a difference of squares. It factors into \((x-3)(x+3)\).

Assess the second denominator, \(x-3\). Here, it already appears in the first denominator's factored form, leaving us to include the additional factor \((x+3)\) to match. That's how a common denominator of \((x-3)(x+3)\) is determined.

When fractions share a common denominator, algebraic operations like addition, subtraction, and simplification become possible. This step is foundational in successful algebraic simplification.

Factoring

Factoring plays a big role in algebraic simplification. It allows us to break down complex expressions into simpler parts. Let’s consider the expression with denominators \(x^2 - 9\) and \(x-3\). The first task is factoring \(x^2 - 9\), a difference of squares.

A difference of squares is an expression like \(a^2 - b^2\), which factors into \((a+b)(a-b)\).

• Here, \(x^2 - 9\) represents \(a^2 - b^2\) with \(a = x\) and \(b = 3\).
• So, it becomes \((x+3)(x-3)\).

With the numerator \(5x^2 + 16x\) in the expression \(\frac{x(5x + 16)}{(x-3)(x+3)}\), factoring requires finding common factors.

• We can factor out \(x\), giving \(x(5x + 16)\).

Factoring simplifies expressions and is a key step in uncovering potential simplifications in both numerators and denominators.

Combining Fractions

Once each fraction has a common denominator, the path to combining them opens up. Consider \(\frac{x}{(x-3)(x+3)} + \frac{5x(x+3)}{(x-3)(x+3)}\). Both share the common denominator \((x-3)(x+3)\).

Combine the fractions by adding or subtracting the numerators.

• Add the numerators to create \(x + 5x(x+3)\).
• Simplifying this involves expanding \(5x(x+3)\) to \(5x^2 + 15x\).
• Combine like terms: \(x + 5x^2 + 15x\) becomes \(5x^2 + 16x\).

Combining fractions results in a single simplified fraction. This method is essential when solving algebraic expressions with fractions, as it reduces complexity and paves the way for further simplification or solution.