https://r2.erweima.ai/mathbot/image/1725923253331wi7cd65i.jpg

The expression in the image is the subtraction of two fractions:

\[
\frac{2a + 3b}{3a - 2b} - \frac{3a + 2b}{3b - 2a}
\]

To simplify or solve this subtraction, we need to follow these steps:

### Step 1: Find a common denominator
To subtract the two fractions, we need a common denominator, which would be the product of the two denominators. So the common denominator would be:

\[
(3a - 2b)(3b - 2a)
\]

### Step 2: Rewrite the fractions with the common denominator
Now, we rewrite each fraction with the common denominator:

\[
\frac{(2a + 3b)(3b - 2a)}{(3a - 2b)(3b - 2a)} - \frac{(3a + 2b)(3a - 2b)}{(3a - 2b)(3b - 2a)}
\]

### Step 3: Combine the numerators
Now, we subtract the numerators:

\[
\frac{(2a + 3b)(3b - 2a) - (3a + 2b)(3a - 2b)}{(3a - 2b)(3b - 2a)}
\]

### Step 4: Expand the numerators
We'll expand each product in the numerator.

1. Expand \((2a + 3b)(3b - 2a)\):

\[
(2a + 3b)(3b - 2a) = 2a(3b) + 2a(-2a) + 3b(3b) + 3b(-2a)
\]
\[
= 6ab - 4a^2 + 9b^2 - 6ab = -4a^2 + 9b^2
\]

2. Expand \((3a + 2b)(3a - 2b)\):

\[
(3a + 2b)(3a - 2b) = (3a)^2 - (2b)^2
\]
\[
= 9a^2 - 4b^2
\]

### Step 5: Combine the expanded numerators
Now we can write the expression for the numerator:

\[
\frac{-4a^2 + 9b^2 - (9a^2 - 4b^2)}{(3a - 2b)(3b - 2a)}
\]

Distribute the negative sign:

\[
= \frac{-4a^2 + 9b^2 - 9a^2 + 4b^2}{(3a - 2b)(3b - 2a)}
\]
\[
= \frac{-4a^2 - 9a^2 + 9b^2 + 4b^2}{(3a - 2b)(3b - 2a)}
\]
\[
= \frac{-13a^2 + 13b^2}{(3a - 2b)(3b - 2a)}
\]

### Step 6: Factor the numerator
Factor out \(13\) from the numerator:

\[
= \frac{13(-a^2 + b^2)}{(3a - 2b)(3b - 2a)}
\]

We know that \( -a^2 + b^2 = -(a^2 - b^2) \), so we can write:

\[
= \frac{13(-(a^2 - b^2))}{(3a - 2b)(3b - 2a)}
\]
\[
= \frac{-13(a^2 - b^2)}{(3a - 2b)(3b - 2a)}
\]

### Final Answer:
The simplified expression is:

\[
\boxed{\frac{-13(a^2 - b^2)}{(3a - 2b)(3b - 2a)}}
\]

Would you like more details on this process or have any other questions?

### Related Questions:
1. How do you handle the addition of rational expressions with different denominators?
2. What are the common techniques for factoring quadratic expressions like \(a^2 - b^2\)?
3. How do you solve equations involving rational expressions with variables in the denominators?
4. Can you simplify the difference of squares for higher powers like \(a^4 - b^4\)?
5. How do you find the least common denominator for multiple rational expressions?

### Tip:
Always remember to factor the numerators and denominators whenever possible—it can simplify expressions quickly!

Learn how to subtract algebraic fractions with different denominators using the common denominator method. This problem covers concepts such as factoring, difference of squares, and algebraic simplification, 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