https://r2.erweima.ai/mathbot/image/1725854641845jm310n01.jpg

Let's simplify the given expression step-by-step. The expression to simplify is:

\[
S = \frac{x^2 - 5x + 6}{x^2 + 2x - 8}
\]

### Step 1: Factor the numerator and the denominator.

#### Numerator: \( x^2 - 5x + 6 \)
We need to find two numbers that multiply to 6 (the constant term) and add up to -5 (the coefficient of \(x\)).

Those numbers are -2 and -3, so we can factor the numerator as:

\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

#### Denominator: \( x^2 + 2x - 8 \)
Similarly, we need two numbers that multiply to -8 and add to 2.

Those numbers are 4 and -2, so we factor the denominator as:

\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\]

### Step 2: Simplify the expression.

Now the expression becomes:

\[
S = \frac{(x - 2)(x - 3)}{(x + 4)(x - 2)}
\]

We can cancel out the common factor \(x - 2\) from both the numerator and the denominator (assuming \(x \neq 2\) to avoid division by zero):

\[
S = \frac{x - 3}{x + 4}
\]

### Final Simplified Expression:

\[
S = \frac{x - 3}{x + 4}
\]

Thus, the correct answer is **A**: \( \frac{x - 3}{x + 4} \).

---

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

Here are 5 related questions to explore this further:
1. How do we handle factoring quadratics when the leading coefficient is not 1?
2. What happens when there are no common factors to cancel between the numerator and the denominator?
3. Can we always cancel terms in rational expressions like this?
4. How do we find restrictions for \(x\) in rational expressions?
5. What methods can be used to simplify more complex rational expressions?

**Tip:** Always check for restrictions in the domain of rational expressions by determining the values of \(x\) that make the denominator zero.

Simplify: S = (x^2 - 5x + 6) / (x^2 + 2x - 8).

Learn how to simplify the rational expression (x^2 - 5x + 6) / (x^2 + 2x - 8) by factoring and canceling common terms. This problem explores algebraic concepts and theorems related to quadratic equations, suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation