Simplify: $\frac{2}{x^2-9}-\frac{1}{x^2-3x}$

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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 focus on simplifying the complex fraction 2x2−9−1x2−3x\frac{2}{x^2-9}-\frac{1}{x^2-3x}, exploring various methods and strategies to achieve this goal.

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: 2x2−9\frac{2}{x^2-9} and 1x2−3x\frac{1}{x^2-3x}. To simplify this expression, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

The first step in simplifying the 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 x2−9x^2-9 and x2−3xx^2-3x. To find the LCM, we can factor each denominator:

x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3)

x2−3x=x(x−3)x^2-3x = x(x-3)

The LCM of (x−3)(x+3)(x-3)(x+3) and x(x−3)x(x-3) is (x−3)(x+3)(x-3)(x+3), since x(x−3)x(x-3) is a factor of (x−3)(x+3)(x-3)(x+3).

Simplifying the Complex Fraction

Now that we have found the common denominator, we can rewrite the complex fraction with the common denominator:

2x2−9−1x2−3x=2(x−3)(x−3)(x+3)−1(x+3)(x−3)(x+3)\frac{2}{x^2-9}-\frac{1}{x^2-3x} = \frac{2(x-3)}{(x-3)(x+3)}-\frac{1(x+3)}{(x-3)(x+3)}

Combining the Fractions

The next step is to combine the fractions by adding or subtracting the numerators:

2(x−3)(x−3)(x+3)−1(x+3)(x−3)(x+3)=2(x−3)−1(x+3)(x−3)(x+3)\frac{2(x-3)}{(x-3)(x+3)}-\frac{1(x+3)}{(x-3)(x+3)} = \frac{2(x-3)-1(x+3)}{(x-3)(x+3)}

Simplifying the Numerator

Now, we can simplify the numerator by combining like terms:

2(x−3)−1(x+3)=2x−6−x−3=x−92(x-3)-1(x+3) = 2x-6-x-3 = x-9

Final Simplification

The final step is to simplify the complex fraction by canceling out any common factors between the numerator and denominator:

x−9(x−3)(x+3)\frac{x-9}{(x-3)(x+3)}

Alternative Method: Using the Least Common Multiple (LCM)

Another approach to simplifying the complex fraction is to use the LCM of the denominators. In this case, the LCM of x2−9x^2-9 and x2−3xx^2-3x is (x−3)(x+3)(x-3)(x+3). We can rewrite the complex fraction with the LCM as the denominator:

2x2−9−1x2−3x=2(x+3)(x−3)(x+3)−1(x−3)(x−3)(x+3)\frac{2}{x^2-9}-\frac{1}{x^2-3x} = \frac{2(x+3)}{(x-3)(x+3)}-\frac{1(x-3)}{(x-3)(x+3)}

Combining the Fractions

The next step is to combine the fractions by adding or subtracting the numerators:

2(x+3)(x−3)(x+3)−1(x−3)(x−3)(x+3)=2(x+3)−1(x−3)(x−3)(x+3)\frac{2(x+3)}{(x-3)(x+3)}-\frac{1(x-3)}{(x-3)(x+3)} = \frac{2(x+3)-1(x-3)}{(x-3)(x+3)}

Simplifying the Numerator

Now, we can simplify the numerator by combining like terms:

2(x+3)−1(x−3)=2x+6−x+3=x+92(x+3)-1(x-3) = 2x+6-x+3 = x+9

Final Simplification

The final step is to simplify the complex fraction by canceling out any common factors between the numerator and denominator:

x+9(x−3)(x+3)\frac{x+9}{(x-3)(x+3)}

Conclusion

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 have explored two methods for simplifying the complex fraction 2x2−9−1x2−3x\frac{2}{x^2-9}-\frac{1}{x^2-3x}. The first method involves finding a common denominator and then combining the fractions, while the second method involves using the LCM of the denominators. Both methods result in the same simplified expression: x−9(x−3)(x+3)\frac{x-9}{(x-3)(x+3)} or x+9(x−3)(x+3)\frac{x+9}{(x-3)(x+3)}.

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Introduction

In our previous article, we explored the process of simplifying complex fractions, focusing on the expression 2x2−9−1x2−3x\frac{2}{x^2-9}-\frac{1}{x^2-3x}. We discussed two methods for simplifying this expression: finding a common denominator and using the least common multiple (LCM) of the denominators. In this article, we will address some common questions and concerns related to simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

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 fractions. Alternatively, you can use the LCM of the denominators to simplify the expression.

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

A: The LCM of two fractions is the least common multiple of their denominators.

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

A: To find the LCM of two fractions, you need to factor each denominator and then find the least common multiple of the factors.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors between the numerator and denominator.

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

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

  • Not finding a common denominator
  • Not combining the fractions correctly
  • Not canceling out common factors
  • Not checking for any remaining factors in the numerator or denominator

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 calculator is giving you the correct answer.

Q: How do I know if a complex fraction is already simplified?

A: To determine if a complex fraction is already simplified, you need to check if the numerator and denominator have any common factors. If they do, you can simplify the fraction by canceling out those factors.

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

A: Yes, you can simplify a complex fraction with variables in the denominator. However, you need to be careful when canceling out common factors, as the variables may not cancel out in the same way as numerical values.

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

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

  • Calculating probabilities and statistics
  • Solving systems of equations
  • Modeling population growth and decay
  • Analyzing financial data

Conclusion

Simplifying complex fractions can be a challenging task, but with practice and patience, you can become proficient in simplifying even the most complex expressions. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex fractions with ease. Remember to always check your work by hand and to use a calculator only as a tool to verify your answers.