To simplify the expression \(\frac{x^2 - 9}{x^2 - 7x + 12}\), we can begin by factoring both the numerator and the denominator.

### Step 1: Factor the numerator
The numerator is \(x^2 - 9\). This is a difference of squares, so it factors as:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Step 2: Factor the denominator
The denominator is \(x^2 - 7x + 12\). To factor this quadratic expression, we need two numbers that multiply to give 12 and add to give -7. These numbers are -3 and -4, so we can factor the denominator as:
\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\]

### Step 3: Simplify the expression
Now, substitute the factored forms of the numerator and denominator into the original expression:
\[
\frac{(x - 3)(x + 3)}{(x - 3)(x - 4)}
\]
Since \((x - 3)\) appears in both the numerator and denominator, we can cancel it out (as long as \(x \neq 3\), to avoid division by zero). This gives us the simplified expression:
\[
\frac{x + 3}{x - 4}
\]

### Final Answer:
\[
\frac{x + 3}{x - 4}, \quad \text{where} \quad x \neq 3 \quad \text{and} \quad x \neq 4 \, (\text{to avoid division by zero}).
\]

Would you like more details on any step?

Here are 5 related questions:
1. How do you factor a quadratic expression like \(x^2 - 7x + 12\)?
2. What is the difference of squares and how do you recognize it?
3. Why do we have to exclude \(x = 3\) and \(x = 4\) from the solution?
4. How do you solve rational expressions in general?
5. What happens when you cannot cancel any factors in a rational expression?

**Tip:** Always check for excluded values after canceling terms in rational expressions to avoid dividing by zero!

Simplify the rational expression x²-9/x²-7x+12 by factoring both the numerator and denominator. This problem explores key algebraic techniques such as factoring quadratics and the difference of squares. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation