Simplify the rational expression 
x
2
−
9
x
2
−
x
−
6

We are given the rational expression:

\[
\frac{x^2 - 9}{x^2 - x - 6}
\]

### Step 1: Factor the numerator and denominator

#### Numerator: \( x^2 - 9 \)
This is a difference of squares, and it can be factored as:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

#### Denominator: \( x^2 - x - 6 \)
We need to find two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \(x\)). These numbers are \(-3\) and \(2\). So, the denominator factors as:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### Step 2: Rewrite the rational expression

Now, substitute the factored forms of the numerator and denominator:

\[
\frac{x^2 - 9}{x^2 - x - 6} = \frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}
\]

### Step 3: Cancel out the common factor

Both the numerator and denominator have a common factor of \( (x - 3) \). We can cancel this factor, assuming \( x \neq 3 \) (since division by zero is undefined):

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

### Final Answer:
The simplified expression is:

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

**Note:** This simplification is valid only for \( x \neq 3 \), since \( x = 3 \) would make the original expression undefined.

Would you like any additional details or clarifications?

Here are some related questions you can explore:
1. How do you factor quadratic expressions like \(x^2 - x - 6\)?
2. What happens when you try to simplify expressions with different denominators?
3. How does the concept of "difference of squares" help in factoring expressions?
4. What is the meaning of canceling common factors in rational expressions?
5. How do you deal with restrictions in rational expressions?

**Tip:** Always check for restrictions in rational expressions after simplifying. These restrictions are values that make the denominator zero, as division by zero is undefined.

Simplify the rational expression x^2 − 9 / x^2 − x − 6

Learn how to simplify the rational expression (x^2 − 9) / (x^2 − x − 6) step-by-step. This solution covers factoring using the difference of squares and quadratic trinomial methods, leading to a simplified form of (x + 3) / (x + 2) with restrictions on x. Suitable for Grades 9-10.

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