Add The Following Expressions:$\[ \frac{2}{3x^2 - 4x - 7} + \frac{1}{3x^2 - 19x + 28} \\]Simplify Your Answer As Much As Possible.

Introduction

In mathematics, complex fractions are a type of expression that involves multiple fractions within a single fraction. These expressions can be challenging to simplify, but with the right approach, they can be broken down into more manageable parts. In this article, we will explore how to simplify complex fractions, using the expression $\frac{2}{3x^2 - 4x - 7} + \frac{1}{3x^2 - 19x + 28}$ as a case study.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given expression, we have two fractions: $\frac{2}{3x^2 - 4x - 7}$ and $\frac{1}{3x^2 - 19x + 28}$. To simplify this expression, we need to find a common denominator and then combine the two fractions.

Finding a Common Denominator

The first step in simplifying a complex fraction is to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are $3x^2 - 4x - 7$ and $3x^2 - 19x + 28$. To find the LCM, we need to factor each denominator.

Factoring the Denominators

• $3x^2 - 4x - 7$ can be factored as $(3x + 7)(x - 1)$.
• $3x^2 - 19x + 28$ can be factored as $(3x - 7)(x - 4)$.

Finding the LCM

The LCM of $(3x + 7)(x - 1)$ and $(3x - 7)(x - 4)$ is $(3x + 7)(x - 1)(3x - 7)(x - 4)$.

Simplifying the Complex Fraction

Now that we have found the common denominator, we can rewrite the expression as:

$$\frac{2(x - 4)(3x - 7)}{(3x + 7)(x - 1)(3x - 7)(x - 4)} + \frac{1(x - 1)(3x + 7)}{(3x + 7)(x - 1)(3x - 7)(x - 4)}$$

We can simplify this expression by canceling out common factors in the numerator and denominator.

Canceling Out Common Factors

• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(x - 4)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(x - 1)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{2}{(3x + 7)(3x - 7)} + \frac{1}{(3x + 7)(3x - 7)}$$

Combining the Fractions

Now that we have a common denominator, we can combine the two fractions.

$$\frac{2 + 1}{(3x + 7)(3x - 7)}$$

Simplifying the Expression

The expression can be simplified by combining the numerators.

$$\frac{3}{(3x + 7)(3x - 7)}$$

Factoring the Denominator

The denominator can be factored as $(3x + 7)(3x - 7) = 9x^2 - 49$.

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{9x^2 - 49}$$

Factoring the Numerator

The numerator can be factored as $3 = 3(1)$.

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3(1)}{9x^2 - 49}$$

Factoring the Denominator

The denominator can be factored as $9x^2 - 49 = (3x - 7)(3x + 7)$.

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3(1)}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{(3x + 7)(3x - 7)}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{9x^2 - 49}$$

Factoring the Denominator

The denominator can be factored as $9x^2 - 49 = (3x - 7)(3x + 7)$.

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

The expression can be simplified by rewriting it as:

$$\frac{3}{(3x - 7)(3x + 7)}$$

Canceling Out Common Factors

• $(3x + 7)$ is a common factor in the numerator and denominator, so we can cancel it out.
• $(3x - 7)$ is a common factor in the numerator and denominator, so we can cancel it out.

After canceling out the common factors, the expression becomes:

$$\frac{3}{9x^2 - 49}$$

Simplifying the Expression

**Simplifying Complex Fractions: A Step-by-Step Guide** =====================================================

Q&A: Simplifying Complex Fractions

Q: What is a complex fraction?

A: A complex fraction is a type of expression that involves multiple fractions within a single fraction. It can be challenging to simplify, but with the right approach, it can be broken down into more manageable parts.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to find a common denominator and then combine the two fractions. The common denominator is the least common multiple (LCM) of the denominators of the two fractions.

Q: What is the least common multiple (LCM)?

A: The LCM is the smallest multiple that both numbers have in common. For example, the LCM of 6 and 8 is 24.

Q: How do I find the LCM of two fractions?

A: To find the LCM of two fractions, you need to factor each fraction and then multiply the factors together.

Q: What is factoring?

A: Factoring is the process of breaking down a number or expression into its prime factors.

Q: How do I factor a fraction?

A: To factor a fraction, you need to factor the numerator and denominator separately and then multiply the factors together.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number of a fraction, and the denominator is the bottom number.

Q: How do I simplify a complex fraction with a common denominator?

A: To simplify a complex fraction with a common denominator, you need to combine the two fractions by adding or subtracting the numerators.

Q: What is the final answer to the complex fraction $\frac{2}{3x^2 - 4x - 7} + \frac{1}{3x^2 - 19x + 28}$?

A: The final answer is $\frac{3}{9x^2 - 49}$.

Q: Can I simplify a complex fraction with a variable in the denominator?

A: Yes, you can simplify a complex fraction with a variable in the denominator by factoring the denominator and then canceling out common factors.

Q: What is the final answer to the complex fraction $\frac{2x}{x^2 - 4x - 7} + \frac{x}{x^2 - 19x + 28}$?

A: The final answer is $\frac{3x}{(x - 4)(x - 7)}$.

Q: Can I simplify a complex fraction with a negative exponent?

A: Yes, you can simplify a complex fraction with a negative exponent by rewriting the fraction with a positive exponent and then simplifying.

Q: What is the final answer to the complex fraction $\frac{2}{x^2 - 4x - 7} - \frac{1}{x^2 - 19x + 28}$?

A: The final answer is $\frac{1}{(x - 4)(x - 7)}$.

Conclusion

Simplifying complex fractions can be challenging, but with the right approach, it can be broken down into more manageable parts. By finding a common denominator and then combining the two fractions, you can simplify complex fractions and arrive at a final answer. Remember to factor the denominator and cancel out common factors to simplify the expression.