https://file.aiquickdraw.com/mathbot/file/17264372350884sryi0ce.jpg

Let's work through this problem together.

### Expression:
\[
\frac{3}{5x^2} + \frac{1}{2x^2 + 6x}
\]

### Part A: State the Least Common Denominator (LCD)
The denominators of the fractions are \(5x^2\) and \(2x^2 + 6x\).

1. **Factor the denominators**:
   - \(5x^2\) is already factored.
   - \(2x^2 + 6x = 2x(x + 3)\).

2. **Find the Least Common Denominator (LCD)**:
   The LCD must include all factors from both denominators, taking the highest powers of common factors.
   - From \(5x^2\), we get \(5x^2\).
   - From \(2x(x + 3)\), we get \(2x(x + 3)\).

   The LCD is the product of these factors:
   \[
   \text{LCD} = 10x^2(x + 3)
   \]

### Part B: Explain how you found the Least Common Denominator
To find the LCD:
- First, factor each denominator:
  - \(5x^2\) remains as it is.
  - \(2x^2 + 6x\) can be factored as \(2x(x + 3)\).
- Then, determine the LCD by taking the least common multiple of all the factors. Since the factor \(x^2\) is common in both denominators, we take the highest power of \(x\), which is \(x^2\). The constants 5 and 2 combine to form the multiple \(10\). The factor \(x + 3\) appears in one of the denominators, so it must be included.

Thus, the LCD is \(10x^2(x + 3)\).

---

Do you need any more details on this, or any other questions? Here are some related questions to further explore:

1. How do you simplify expressions with different denominators once you find the LCD?
2. How would you approach adding fractions if the denominators cannot be factored?
3. What would happen if the denominators had no common factors? 
4. How does factoring polynomials help in finding the LCD of rational expressions?
5. What are some common mistakes to avoid when finding the LCD of algebraic fractions?

**Tip:** Always factor denominators completely before attempting to find the LCD. This will help you avoid missing common factors.

Consider the following expression: 3/(5x^2) + 1/(2x^2 + 6x). A) State the Least Common Denominator (LCD) for the expression. B) Explain how you found the Least Common Denominator in Part A.

Learn how to find the Least Common Denominator (LCD) for the expression 3/(5x^2) + 1/(2x^2 + 6x) through factoring and multiplying the unique factors. Step-by-step explanation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation