Simplify The Expression:${ \frac{7x+2}{x^2-7x+10} - \frac{x}{x-5} }$

by ADMIN 70 views

Introduction

Simplifying complex fractions can be a daunting task, especially when dealing with expressions involving polynomials and rational functions. In this article, we will focus on simplifying the given expression: 7x+2x2−7x+10−xx−5\frac{7x+2}{x^2-7x+10} - \frac{x}{x-5}. We will break down the solution into manageable steps, using algebraic techniques and mathematical concepts to simplify the expression.

Understanding the Expression

The given expression involves two fractions: 7x+2x2−7x+10\frac{7x+2}{x^2-7x+10} and xx−5\frac{x}{x-5}. To simplify the expression, we need to find a common denominator and combine the fractions. However, before we can do that, we need to factor the denominators and simplify the expressions.

Factoring the Denominators

The first denominator, x2−7x+10x^2-7x+10, can be factored as (x−5)(x−2)(x-5)(x-2). The second denominator, x−5x-5, is already in its simplest form.

Simplifying the Expressions

Now that we have factored the denominators, we can simplify the expressions. The first fraction can be rewritten as 7x+2(x−5)(x−2)\frac{7x+2}{(x-5)(x-2)}. The second fraction remains the same: xx−5\frac{x}{x-5}.

Finding a Common Denominator

To combine the fractions, we need to find a common denominator. In this case, the common denominator is (x−5)(x−2)(x-5)(x-2). We can rewrite the second fraction with the common denominator: x(x−2)(x−5)(x−2)\frac{x(x-2)}{(x-5)(x-2)}.

Combining the Fractions

Now that we have a common denominator, we can combine the fractions. We can rewrite the expression as 7x+2(x−5)(x−2)−x(x−2)(x−5)(x−2)\frac{7x+2}{(x-5)(x-2)} - \frac{x(x-2)}{(x-5)(x-2)}.

Simplifying the Expression

To simplify the expression, we can subtract the numerators: 7x+2−x(x−2)(x−5)(x−2)\frac{7x+2 - x(x-2)}{(x-5)(x-2)}. We can simplify the numerator by expanding and combining like terms: 7x+2−x2+2x(x−5)(x−2)\frac{7x+2 - x^2 + 2x}{(x-5)(x-2)}.

Final Simplification

We can simplify the numerator further by combining like terms: −x2+9x+2(x−5)(x−2)\frac{-x^2 + 9x + 2}{(x-5)(x-2)}.

Conclusion

Simplifying complex fractions requires a step-by-step approach, using algebraic techniques and mathematical concepts. By factoring the denominators, simplifying the expressions, finding a common denominator, and combining the fractions, we can simplify the given expression: 7x+2x2−7x+10−xx−5\frac{7x+2}{x^2-7x+10} - \frac{x}{x-5}. The final simplified expression is −x2+9x+2(x−5)(x−2)\frac{-x^2 + 9x + 2}{(x-5)(x-2)}.

Additional Tips and Tricks

  • When simplifying complex fractions, it's essential to factor the denominators and simplify the expressions.
  • Finding a common denominator is crucial in combining fractions.
  • Be careful when expanding and combining like terms in the numerator.
  • Simplifying complex fractions requires patience and attention to detail.

Real-World Applications

Simplifying complex fractions has numerous real-world applications in various fields, including:

  • Physics: Simplifying complex fractions is essential in solving problems involving motion, energy, and momentum.
  • Engineering: Simplifying complex fractions is crucial in designing and analyzing complex systems, such as electrical circuits and mechanical systems.
  • Economics: Simplifying complex fractions is essential in modeling and analyzing economic systems, including supply and demand curves.

Final Thoughts

Simplifying complex fractions may seem daunting at first, but with practice and patience, it becomes a manageable task. By following the step-by-step approach outlined in this article, you can simplify even the most complex fractions. Remember to factor the denominators, simplify the expressions, find a common denominator, and combine the fractions. With these techniques, you'll be able to simplify complex fractions with ease.

Introduction

Simplifying complex fractions can be a challenging task, but with the right techniques and strategies, it becomes more manageable. In this article, we will provide a Q&A guide to help you understand and simplify complex fractions. We will cover common questions and concerns, and provide step-by-step solutions to help you master the art of simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. It can be a simple fraction, such as 12\frac{1}{2}, or a more complex fraction, such as 11+x\frac{1}{1+x}.

Q: How do I simplify a complex fraction?

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

  1. Factor the denominators and simplify the expressions.
  2. Find a common denominator.
  3. Combine the fractions.
  4. Simplify the resulting expression.

Q: What is the difference between a complex fraction and a rational expression?

A: A rational expression is a fraction that contains polynomials in its numerator and denominator. A complex fraction, on the other hand, is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

Q: What is the difference between a complex fraction and an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables and constants. A complex fraction, on the other hand, is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

  1. Combine like terms.
  2. Simplify the resulting expression.

Q: What is the most common mistake when simplifying complex fractions?

A: The most common mistake when simplifying complex fractions is to forget to factor the denominators and simplify the expressions.

Q: How do I avoid making this mistake?

A: To avoid making this mistake, you need to carefully read and follow the steps outlined in the solution. You should also double-check your work to ensure that you have factored the denominators and simplified the expressions correctly.

Q: What is the best way to practice simplifying complex fractions?

A: The best way to practice simplifying complex fractions is to work through a series of examples and exercises. You can find these in textbooks, online resources, or by creating your own practice problems.

Q: How do I know if I have simplified a complex fraction correctly?

A: To know if you have simplified a complex fraction correctly, you need to check your work carefully. You should also compare your solution to the original expression to ensure that you have obtained the correct result.

Q: What are some common applications of simplifying complex fractions?

A: Simplifying complex fractions has numerous real-world applications in various fields, including:

  • Physics: Simplifying complex fractions is essential in solving problems involving motion, energy, and momentum.
  • Engineering: Simplifying complex fractions is crucial in designing and analyzing complex systems, such as electrical circuits and mechanical systems.
  • Economics: Simplifying complex fractions is essential in modeling and analyzing economic systems, including supply and demand curves.

Conclusion

Simplifying complex fractions requires a step-by-step approach, using algebraic techniques and mathematical concepts. By following the Q&A guide outlined in this article, you can master the art of simplifying complex fractions and apply it to real-world problems. Remember to factor the denominators, simplify the expressions, find a common denominator, and combine the fractions. With practice and patience, you'll be able to simplify complex fractions with ease.