Find The LCD For \[$\frac{6}{4w^2 + 23w + 28}, \frac{10w}{4w^2 - W - 14}\$\]

$$

Introduction

In mathematics, finding the least common denominator (LCD) is a crucial step in adding, subtracting, multiplying, or dividing fractions. The LCD is the smallest multiple that all the denominators of the fractions have in common. In this article, we will focus on finding the LCD for two given fractions, $\frac{6}{4w^2 + 23w + 28}$ and $\frac{10w}{4w^2 - w - 14}$.

Understanding the Problem

To find the LCD, we need to first factorize the denominators of both fractions. The first fraction has a denominator of $4w^2 + 23w + 28$, and the second fraction has a denominator of $4w^2 - w - 14$. We will start by factorizing the first denominator.

Factorizing the First Denominator

The first denominator is $4w^2 + 23w + 28$. To factorize this quadratic expression, we need to find two numbers whose product is $4 \times 28 = 112$ and whose sum is $23$. These numbers are $16$ and $7$, since $16 \times 7 = 112$ and $16 + 7 = 23$. Therefore, we can write the first denominator as $(4w + 16)(w + 7)$.

Factorizing the Second Denominator

The second denominator is $4w^2 - w - 14$. To factorize this quadratic expression, we need to find two numbers whose product is $4 \times -14 = -56$ and whose sum is $-1$. These numbers are $-8$ and $7$, since $-8 \times 7 = -56$ and $-8 + 7 = -1$. Therefore, we can write the second denominator as $(4w - 8)(w + 7)$.

Finding the LCD

Now that we have factorized both denominators, we can find the LCD by taking the product of all the factors. The first denominator is $(4w + 16)(w + 7)$, and the second denominator is $(4w - 8)(w + 7)$. Since both denominators have a common factor of $(w + 7)$, we can cancel this factor out to get the LCD.

Canceling Common Factors

To cancel the common factor of $(w + 7)$, we need to multiply both the numerator and the denominator of each fraction by the reciprocal of the common factor. For the first fraction, we multiply the numerator and denominator by $\frac{1}{w + 7}$, and for the second fraction, we multiply the numerator and denominator by $\frac{1}{w + 7}$.

Simplifying the Fractions

After canceling the common factor of $(w + 7)$, we can simplify the fractions. The first fraction becomes $\frac{6}{4w + 16}$, and the second fraction becomes $\frac{10w}{4w - 8}$.

Conclusion

In conclusion, the LCD for the given fractions is the product of all the factors, which is $(4w + 16)(w + 7)(4w - 8)$. We can find the LCD by factorizing the denominators, canceling common factors, and simplifying the fractions. This is a crucial step in adding, subtracting, multiplying, or dividing fractions, and it requires a good understanding of algebraic expressions and factorization.

Example Use Case

Suppose we want to add the two fractions $\frac{6}{4w^2 + 23w + 28}$ and $\frac{10w}{4w^2 - w - 14}$. To do this, we need to find the LCD, which is $(4w + 16)(w + 7)(4w - 8)$. We can then rewrite each fraction with the LCD as the denominator and add the numerators.

Rewriting the Fractions

We can rewrite the first fraction as $\frac{6(4w - 8)}{(4w + 16)(w + 7)(4w - 8)}$ and the second fraction as $\frac{10w(w + 7)}{(4w + 16)(w + 7)(4w - 8)}$.

Adding the Fractions

Now that we have rewritten the fractions with the LCD as the denominator, we can add the numerators. The sum of the fractions is $\frac{6(4w - 8) + 10w(w + 7)}{(4w + 16)(w + 7)(4w - 8)}$.

Final Answer

The final answer is $\boxed{(4w + 16)(w + 7)(4w - 8)}$.

$$

Introduction

In our previous article, we discussed how to find the least common denominator (LCD) for two given fractions, $\frac{6}{4w^2 + 23w + 28}$ and $\frac{10w}{4w^2 - w - 14}$. In this article, we will answer some frequently asked questions about finding the LCD.

Q&A

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest multiple that all the denominators of the fractions have in common.

Q: Why is it important to find the LCD?

A: Finding the LCD is crucial when adding, subtracting, multiplying, or dividing fractions. It ensures that the fractions have a common denominator, making it easier to perform operations.

Q: How do I find the LCD?

A: To find the LCD, you need to factorize the denominators of both fractions, cancel common factors, and simplify the fractions.

Q: What if the denominators do not have any common factors?

A: If the denominators do not have any common factors, the LCD is the product of all the factors of each denominator.

Q: Can I use a calculator to find the LCD?

A: Yes, you can use a calculator to find the LCD. However, it's essential to understand the concept and be able to factorize expressions manually.

Q: How do I simplify fractions after finding the LCD?

A: After finding the LCD, you can simplify the fractions by canceling common factors between the numerator and the denominator.

Q: Can I add or subtract fractions with different denominators?

A: No, you cannot add or subtract fractions with different denominators. You need to find the LCD first and then rewrite the fractions with the LCD as the denominator.

Q: What if I have multiple fractions with different denominators?

A: If you have multiple fractions with different denominators, you need to find the LCD for all the fractions and then rewrite each fraction with the LCD as the denominator.

Q: Can I use the LCD to multiply or divide fractions?

A: Yes, you can use the LCD to multiply or divide fractions. However, you need to follow the rules of multiplying or dividing fractions, which involves multiplying or dividing the numerators and denominators separately.

Example Use Cases

Adding Fractions with Different Denominators

Suppose we want to add the fractions $\frac{6}{4w^2 + 23w + 28}$ and $\frac{10w}{4w^2 - w - 14}$. To do this, we need to find the LCD, which is $(4w + 16)(w + 7)(4w - 8)$. We can then rewrite each fraction with the LCD as the denominator and add the numerators.

Multiplying Fractions with Different Denominators

Suppose we want to multiply the fractions $\frac{6}{4w^2 + 23w + 28}$ and $\frac{10w}{4w^2 - w - 14}$. To do this, we need to find the LCD, which is $(4w + 16)(w + 7)(4w - 8)$. We can then rewrite each fraction with the LCD as the denominator and multiply the numerators and denominators separately.

Conclusion

In conclusion, finding the least common denominator (LCD) is a crucial step in adding, subtracting, multiplying, or dividing fractions. It ensures that the fractions have a common denominator, making it easier to perform operations. We hope this Q&A article has helped you understand the concept of finding the LCD and how to apply it in different scenarios.

Final Answer

The final answer is $\boxed{(4w + 16)(w + 7)(4w - 8)}$.