The given expression is:

\[
\frac{6x^2 + 5xy - 6y^2}{3x^2 + 7xy - 6y^2}
\]

To simplify this expression, let's follow these steps:

1. **Factor the Numerator** \(6x^2 + 5xy - 6y^2\):
   - We need to find two numbers whose product is \(6x^2 \times (-6y^2) = -36x^2y^2\) and whose sum is \(5xy\).
   - These numbers are \(9xy\) and \(-4xy\), so we rewrite the middle term:

   \[
   6x^2 + 9xy - 4xy - 6y^2
   \]

   - Now, factor by grouping:

   \[
   3x(2x + 3y) - 2y(2x + 3y)
   \]

   - Factor out the common binomial:

   \[
   (3x - 2y)(2x + 3y)
   \]

2. **Factor the Denominator** \(3x^2 + 7xy - 6y^2\):
   - We need to find two numbers whose product is \(3x^2 \times (-6y^2) = -18x^2y^2\) and whose sum is \(7xy\).
   - These numbers are \(9xy\) and \(-2xy\), so we rewrite the middle term:

   \[
   3x^2 + 9xy - 2xy - 6y^2
   \]

   - Now, factor by grouping:

   \[
   3x(x + 3y) - 2y(x + 3y)
   \]

   - Factor out the common binomial:

   \[
   (3x - 2y)(x + 3y)
   \]

3. **Simplify the Expression**:
   - Now substitute the factored forms back into the original expression:

   \[
   \frac{(3x - 2y)(2x + 3y)}{(3x - 2y)(x + 3y)}
   \]

   - Cancel out the common factor \(3x - 2y\):

   \[
   \frac{2x + 3y}{x + 3y}
   \]

This is the simplified form of the given expression.

Would you like further details or have any questions?

---

Here are some related questions you might find helpful:

1. How do you factor trinomials of the form \(ax^2 + bx + c\)?
2. What are the steps to factor by grouping in polynomials?
3. How do you simplify rational expressions?
4. What happens if a term in the numerator and denominator cancels out completely?
5. How can you recognize common binomials when factoring?

**Tip:** When factoring trinomials, always check if there is a common factor in all the terms first before attempting to factor further.

Learn how to simplify the rational expression (6x² + 5xy - 6y²) / (3x² + 7xy - 6y²) step-by-step. Factorization techniques and algebraic manipulations are used to simplify this expression, covering algebraic concepts such as rational expressions and factoring. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation