Math Problem Statement

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

Solution

We are given the rational expression:

x29x2x6\frac{x^2 - 9}{x^2 - x - 6}

Step 1: Factor the numerator and denominator

Numerator: x29x^2 - 9

This is a difference of squares, and it can be factored as:

x29=(x3)(x+3)x^2 - 9 = (x - 3)(x + 3)

Denominator: x2x6x^2 - x - 6

We need to find two numbers that multiply to 6-6 (the constant term) and add to 1-1 (the coefficient of xx). These numbers are 3-3 and 22. So, the denominator factors as:

x2x6=(x3)(x+2)x^2 - x - 6 = (x - 3)(x + 2)

Step 2: Rewrite the rational expression

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

x29x2x6=(x3)(x+3)(x3)(x+2)\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 (x3)(x - 3). We can cancel this factor, assuming x3x \neq 3 (since division by zero is undefined):

(x3)(x+3)(x3)(x+2)=x+3x+2\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}

Final Answer:

The simplified expression is:

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

Note: This simplification is valid only for x3x \neq 3, since x=3x = 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 x2x6x^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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)
Factoring quadratic trinomials: ax^2 + bx + c

Theorems

Factoring Theorem
Cancellation Law for Rational Expressions

Suitable Grade Level

Grades 9-10