To solve the rational equation \(\frac{3x}{x + 1} = \frac{12}{x^2 - 1} + 2\), we need to follow these steps:

### Step-by-Step Solution

1. **Identify the common denominator:**

   The denominators in the equation are \(x + 1\) and \(x^2 - 1\). Notice that \(x^2 - 1\) can be factored as \((x + 1)(x - 1)\). Thus, the common denominator for all terms is \((x + 1)(x - 1)\).

2. **Rewrite the equation with a common denominator:**

   We need to express each term using the common denominator \((x + 1)(x - 1)\):
   \[
   \frac{3x}{x + 1} = \frac{12}{(x + 1)(x - 1)} + \frac{2(x + 1)(x - 1)}{(x + 1)(x - 1)}.
   \]

   Simplifying the second term:
   \[
   \frac{3x(x - 1)}{(x + 1)(x - 1)} = \frac{12}{(x + 1)(x - 1)} + \frac{2(x^2 - 1)}{(x + 1)(x - 1)}.
   \]

3. **Combine all terms over the common denominator:**

   Now, the equation becomes:
   \[
   \frac{3x(x - 1)}{(x + 1)(x - 1)} = \frac{12 + 2(x^2 - 1)}{(x + 1)(x - 1)}.
   \]

4. **Simplify the numerators:**

   Simplify each side:
   \[
   3x(x - 1) = 12 + 2(x^2 - 1).
   \]
   Expand and simplify:
   \[
   3x^2 - 3x = 12 + 2x^2 - 2.
   \]
   \[
   3x^2 - 3x = 2x^2 + 10.
   \]

5. **Move all terms to one side:**

   Subtract \(2x^2\) from both sides:
   \[
   3x^2 - 2x^2 - 3x - 10 = 0.
   \]
   This simplifies to:
   \[
   x^2 - 3x - 10 = 0.
   \]

6. **Solve the quadratic equation:**

   We can solve \(x^2 - 3x - 10 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -3\), and \(c = -10\):
   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}.
   \]
   \[
   x = \frac{3 \pm \sqrt{9 + 40}}{2}.
   \]
   \[
   x = \frac{3 \pm \sqrt{49}}{2}.
   \]
   \[
   x = \frac{3 \pm 7}{2}.
   \]

   Thus, the solutions are:
   \[
   x = \frac{3 + 7}{2} = 5 \quad \text{and} \quad x = \frac{3 - 7}{2} = -2.
   \]

7. **Check for extraneous solutions:**

   We must check if these solutions make any denominator zero in the original equation:
   - For \(x = 5\), none of the denominators are zero.
   - For \(x = -2\), substituting into \(x + 1 = -1\) and \((x^2 - 1) = 3\), both denominators are not zero.

### Conclusion
Both \(x = 5\) and \(x = -2\) are valid solutions to the equation.

Would you like to ask any further questions or need more details?

Here are some related questions to consider:

1. What if the equation involved higher powers of \(x\)?
2. How do you solve rational equations involving square roots?
3. What strategies can be used to simplify complex rational expressions?
4. How does factoring help in solving rational equations?
5. What are some common pitfalls when solving rational equations?

**Tip:** Always check your solutions in the original equation to ensure they don't make any denominators zero or introduce extraneous solutions.

Learn how to solve the rational equation 3x/(x + 1) = 12/(x^2 - 1) + 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational equations and quadratic equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation