Questions: Add and simplify. x^2-5/6x^2-x-5 + x+9/6x-6 x^2-5/6x^2-x-5 + x+9/6x-6= (Simplify your answer. Factor completely.)

Transcript text: Add and simplify. \[ \begin{array}{l} \frac{x^{2}-5}{6 x^{2}-1 x-5}+\frac{x+9}{6 x-6} \\ \frac{x^{2}-5}{6 x^{2}-1 x-5}+\frac{x+9}{6 x-6}= \end{array} \] $\square$ (Simplify your answer. Factor completely.)

Solution

Solution Steps

To add and simplify the given rational expressions, we need to follow these steps:

Factor the denominators of both fractions.
Find a common denominator.
Rewrite each fraction with the common denominator.
Add the numerators.
Simplify the resulting expression.

Step 1: Factor the Denominators

First, we factor the denominators of both fractions: \[ \text{Denominator 1: } 6x^2 - x - 5 = (x - 1)(6x + 5) \] \[ \text{Denominator 2: } 6x - 6 = 6(x - 1) \]

Step 2: Find a Common Denominator

The common denominator is the least common multiple (LCM) of the factored denominators: \[ \text{LCM}((x - 1)(6x + 5), 6(x - 1)) = 6(x - 1)(6x + 5) \]

Step 3: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction with the common denominator: \[ \frac{x^2 - 5}{(x - 1)(6x + 5)} = \frac{(x^2 - 5) \cdot 6}{6(x - 1)(6x + 5)} \] \[ \frac{x + 9}{6(x - 1)} = \frac{(x + 9) \cdot (6x + 5)}{6(x - 1)(6x + 5)} \]

Step 4: Add the Numerators

Add the numerators of the rewritten fractions: \[ \frac{6(x^2 - 5) + (x + 9)(6x + 5)}{6(x - 1)(6x + 5)} \] Simplify the numerator: \[ 6(x^2 - 5) + (x + 9)(6x + 5) = 6x^2 - 30 + 6x^2 + 59x + 45 = 12x^2 + 59x + 15 \]

Step 5: Simplify the Resulting Expression

Combine the simplified numerator with the common denominator: \[ \frac{12x^2 + 59x + 15}{6(x - 1)(6x + 5)} \]