Simplify x2−4/4x2−12x ÷ 2x+4/−x2+5x−6𝑥

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

\[
\frac{x^2 - 4}{4x^2 - 12x} \div \frac{2x + 4}{-x^2 + 5x - 6}
\]

### Step 1: Simplify each fraction separately

#### Numerator (first fraction):
\[
\frac{x^2 - 4}{4x^2 - 12x}
\]

The numerator \(x^2 - 4\) is a difference of squares, which can be factored as:

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

The denominator \(4x^2 - 12x\) can be factored by taking out the common factor \(4x\):

\[
4x^2 - 12x = 4x(x - 3)
\]

So, the first fraction becomes:

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

#### Denominator (second fraction):
\[
\frac{2x + 4}{-x^2 + 5x - 6}
\]

The numerator \(2x + 4\) can be factored by taking out the common factor \(2\):

\[
2x + 4 = 2(x + 2)
\]

The denominator \(-x^2 + 5x - 6\) can be factored as:

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

So, the second fraction becomes:

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

### Step 2: Perform the division

When dividing by a fraction, multiply by its reciprocal:

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

### Step 3: Simplify the expression

Now, cancel out the common factors:

- \((x + 2)\) appears in the numerator of the first fraction and the denominator of the second fraction, so it cancels out.
- \((x - 2)\) appears in both fractions, so it cancels out.
- \((x - 3)\) also cancels out.

After canceling, we are left with:

\[
\frac{-1}{8x}
\]

### Final Simplified Expression:

\[
\frac{-1}{8x}
\]

This is the simplified form of the given expression.

Would you like more details or have any questions?

Here are 8 related questions that might interest you:
1. What is the difference between multiplying and dividing algebraic fractions?
2. How do you factor polynomials like \(x^2 - 4\)?
3. What is the reciprocal of a fraction, and how is it used in division?
4. How do you identify and cancel common factors in fractions?
5. What are the steps to simplify complex fractions?
6. Can this simplified expression be further simplified if \(x\) is known?
7. What are the rules for factoring quadratic expressions?
8. How do you interpret the negative sign in rational expressions?

**Tip:** When simplifying complex fractions, always factor completely to identify common factors that can be canceled.

Learn how to simplify the expression (x^2 - 4) / (4x^2 - 12x) ÷ (2x + 4) / (-x^2 + 5x - 6𝑥) step-by-step. Includes detailed fraction simplification and factorization techniques. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation