2a-3/a-1=2a+3/a+1 + 10/a^2-1 solve this rational equation

We are tasked with solving the rational equation:

\[
\frac{2a - 3}{a - 1} = \frac{2a + 3}{a + 1} + \frac{10}{a^2 - 1}
\]

### Step 1: Recognize the structure of the equation
The first thing to notice is that \( a^2 - 1 \) is a difference of squares and can be factored as:

\[
a^2 - 1 = (a - 1)(a + 1)
\]

### Step 2: Rewrite the equation
We can now rewrite the equation as:

\[
\frac{2a - 3}{a - 1} = \frac{2a + 3}{a + 1} + \frac{10}{(a - 1)(a + 1)}
\]

### Step 3: Find a common denominator
The common denominator for all terms is \( (a - 1)(a + 1) \).

We can rewrite each term with this denominator:

- The left-hand side becomes:
\[
\frac{2a - 3}{a - 1} = \frac{(2a - 3)(a + 1)}{(a - 1)(a + 1)}
\]

- The right-hand side becomes:
\[
\frac{2a + 3}{a + 1} + \frac{10}{(a - 1)(a + 1)} = \frac{(2a + 3)(a - 1) + 10}{(a - 1)(a + 1)}
\]

### Step 4: Simplify the equation
Now, we have:

\[
\frac{(2a - 3)(a + 1)}{(a - 1)(a + 1)} = \frac{(2a + 3)(a - 1) + 10}{(a - 1)(a + 1)}
\]

Since the denominators are the same, we can equate the numerators:

\[
(2a - 3)(a + 1) = (2a + 3)(a - 1) + 10
\]

### Step 5: Expand both sides
First, expand the left-hand side:

\[
(2a - 3)(a + 1) = 2a^2 + 2a - 3a - 3 = 2a^2 - a - 3
\]

Now, expand the right-hand side:

\[
(2a + 3)(a - 1) = 2a^2 - 2a + 3a - 3 = 2a^2 + a - 3
\]

So, the equation becomes:

\[
2a^2 - a - 3 = 2a^2 + a - 3 + 10
\]

Simplifying the right-hand side:

\[
2a^2 + a - 3 + 10 = 2a^2 + a + 7
\]

### Step 6: Simplify the equation further
Now, subtract \( 2a^2 \) from both sides:

\[
-a - 3 = a + 7
\]

### Step 7: Solve for \( a \)
Move all terms involving \( a \) to one side and constants to the other side:

\[
-a - a = 7 + 3
\]

\[
-2a = 10
\]

\[
a = -5
\]

### Step 8: Check for extraneous solutions
We need to ensure that \( a = -5 \) does not make any denominator zero. The denominators are \( a - 1 \) and \( a + 1 \). For \( a = -5 \):

\[
a - 1 = -5 - 1 = -6, \quad a + 1 = -5 + 1 = -4
\]

Neither of these is zero, so \( a = -5 \) is valid.

### Final Answer:
The solution to the equation is:

\[
a = -5
\]

Would you like further details or explanations on any step?

Here are five follow-up questions to deepen your understanding:
1. How can we factor the difference of squares in other algebraic expressions?
2. What are extraneous solutions and how do they arise in rational equations?
3. How does the common denominator method help in solving rational equations?
4. What other methods can be used to solve rational equations?
5. Why is it important to check for zero denominators in rational equations?

**Tip:** Always factor expressions like \( a^2 - 1 \) to simplify your work early in a problem.

2a - 3/a - 1 = 2a + 3/a + 1 + 10/a^2 - 1 solve this rational equation

This detailed step-by-step guide shows how to solve the rational equation 2a - 3/(a - 1) = 2a + 3/(a + 1) + 10/(a^2 - 1) by using factoring and common denominator methods. Learn how to factor difference of squares and avoid extraneous solutions. Suitable for 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