Add And Simplify: $\[ \frac{x^2 - 9x}{x^2 + 2x - 15} + \frac{9x - 9}{x^2 + 2x - 15} \\]Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box Blank. Answer:-

Introduction

Complex fractions can be a daunting task for many students and mathematicians alike. However, with the right approach and techniques, simplifying these fractions can become a manageable and even enjoyable process. In this article, we will explore the steps involved in simplifying complex fractions, using the given problem 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 problem, we have two fractions:

$$\frac{x^2 - 9x}{x^2 + 2x - 15} + \frac{9x - 9}{x^2 + 2x - 15}$$

The first step in simplifying this fraction is to understand the concept of complex fractions and how they can be simplified.

Step 1: Factor the Denominator

The first step in simplifying the given fraction is to factor the denominator. The denominator is a quadratic expression that can be factored as follows:

$$x^2 + 2x - 15 = (x + 5)(x - 3)$$

By factoring the denominator, we can rewrite the fraction as:

$$\frac{x^2 - 9x}{(x + 5)(x - 3)} + \frac{9x - 9}{(x + 5)(x - 3)}$$

Step 2: Find the Common Denominator

The next step is to find the common denominator of the two fractions. In this case, the common denominator is the factored form of the denominator, which is $(x + 5)(x - 3)$.

Step 3: Combine the Fractions

Now that we have the common denominator, we can combine the two fractions by adding their numerators. The resulting fraction is:

$$\frac{(x^2 - 9x) + (9x - 9)}{(x + 5)(x - 3)}$$

Step 4: Simplify the Numerator

The next step is to simplify the numerator by combining like terms. The resulting fraction is:

$$\frac{x^2 - 9x + 9x - 9}{(x + 5)(x - 3)}$$

Step 5: Cancel Out Common Factors

The final step is to cancel out any common factors between the numerator and denominator. In this case, we can cancel out the common factor of $x$ in the numerator and denominator.

The Final Answer

After simplifying the fraction, we get:

$$\frac{x^2 - 9}{(x + 5)(x - 3)}$$

Conclusion

Simplifying complex fractions requires a step-by-step approach, starting with factoring the denominator, finding the common denominator, combining the fractions, simplifying the numerator, and canceling out common factors. By following these steps, we can simplify even the most complex fractions and arrive at a final answer.

Example Problems

Here are a few example problems to help you practice simplifying complex fractions:

• $\frac{x^2 + 5x}{x^2 - 4x} + \frac{5x - 5}{x^2 - 4x}$
• $\frac{2x^2 - 3x}{x^2 + 2x - 1} + \frac{3x - 3}{x^2 + 2x - 1}$
• $\frac{x^2 + 2x}{x^2 - 3x} + \frac{2x - 2}{x^2 - 3x}$

Tips and Tricks

Here are a few tips and tricks to help you simplify complex fractions:

• Always factor the denominator before simplifying the fraction.
• Find the common denominator before combining the fractions.
• Simplify the numerator by combining like terms.
• Cancel out common factors between the numerator and denominator.

Introduction

Simplifying complex fractions can be a challenging task, but with the right approach and techniques, it can become a manageable and even enjoyable process. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions, using the given problem as a case study.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given problem, we have two fractions:

$$\frac{x^2 - 9x}{x^2 + 2x - 15} + \frac{9x - 9}{x^2 + 2x - 15}$$

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

1. Factor the denominator.
2. Find the common denominator.
3. Combine the fractions.
4. Simplify the numerator.
5. Cancel out common factors.

Q: What is the common denominator?

A: The common denominator is the factored form of the denominator, which is $(x + 5)(x - 3)$ in the given problem.

Q: How do I find the common denominator?

A: To find the common denominator, you need to factor the denominator and identify the common factors between the two fractions.

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. However, you need to be careful when canceling out common factors, as the variable may not be canceled out.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is:

$$\frac{x^2 - 9}{(x + 5)(x - 3)}$$

Q: Can I use a calculator to simplify complex fractions?

A: Yes, you can use a calculator to simplify complex fractions. However, it's always a good idea to check your work by hand to ensure that the answer is correct.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

• Not factoring the denominator.
• Not finding the common denominator.
• Not combining the fractions correctly.
• Not simplifying the numerator.
• Not canceling out common factors.

Q: Can I simplify complex fractions with negative numbers?

A: Yes, you can simplify complex fractions with negative numbers. However, you need to be careful when canceling out common factors, as the negative sign may not be canceled out.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions has many real-world applications, including:

• Calculating interest rates.
• Determining the area of a circle.
• Finding the volume of a sphere.
• Solving systems of equations.

By following these steps and tips, you can simplify even the most complex fractions and arrive at a final answer.

Example Problems

Here are a few example problems to help you practice simplifying complex fractions:

• $\frac{x^2 + 5x}{x^2 - 4x} + \frac{5x - 5}{x^2 - 4x}$
• $\frac{2x^2 - 3x}{x^2 + 2x - 1} + \frac{3x - 3}{x^2 + 2x - 1}$
• $\frac{x^2 + 2x}{x^2 - 3x} + \frac{2x - 2}{x^2 - 3x}$

Tips and Tricks

Here are a few tips and tricks to help you simplify complex fractions:

• Always factor the denominator before simplifying the fraction.
• Find the common denominator before combining the fractions.
• Simplify the numerator by combining like terms.
• Cancel out common factors between the numerator and denominator.

By following these steps and tips, you can simplify even the most complex fractions and arrive at a final answer.