Simplify The Expression:$\[ \frac{x}{x(x+2)} - \frac{2}{(x-1)} \\]

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Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. In this article, we will focus on simplifying the given expression: xx(x+2)−2(x−1)\frac{x}{x(x+2)} - \frac{2}{(x-1)}. We will break down the expression into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Expression

The given expression is a combination of two fractions: xx(x+2)\frac{x}{x(x+2)} and 2(x−1)\frac{2}{(x-1)}. To simplify this expression, we need to find a common denominator, which will allow us to combine the two fractions into a single expression.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are x(x+2)x(x+2) and (x−1)(x-1). We can start by factoring the first denominator:

x(x+2)=x2+2xx(x+2) = x^2 + 2x

Now, we need to find the LCM of x2+2xx^2 + 2x and (x−1)(x-1). To do this, we can list the factors of each expression:

Factors of x2+2xx^2 + 2x: x,x+2,xx, x+2, x

Factors of (x−1)(x-1): x−1x-1

The LCM of x2+2xx^2 + 2x and (x−1)(x-1) is (x−1)(x+2)(x-1)(x+2).

Simplifying the Expression

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

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

Combining the Fractions

Now that the expression has a common denominator, we can combine the two fractions:

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

Expanding the Numerator

To simplify the expression further, we can expand the numerator:

x(x−1)−2(x+2)=x2−x−2x−4x(x-1) - 2(x+2) = x^2 - x - 2x - 4

Combining Like Terms

Now that we have expanded the numerator, we can combine like terms:

x2−x−2x−4=x2−3x−4x^2 - x - 2x - 4 = x^2 - 3x - 4

Simplifying the Expression

Now that we have simplified the numerator, we can rewrite the expression:

x2−3x−4(x−1)(x+2)\frac{x^2 - 3x - 4}{(x-1)(x+2)}

Conclusion

In this article, we have simplified the given expression: xx(x+2)−2(x−1)\frac{x}{x(x+2)} - \frac{2}{(x-1)}. We started by finding a common denominator, which allowed us to combine the two fractions into a single expression. We then expanded the numerator and combined like terms to simplify the expression further. By following these steps, you can simplify complex algebraic expressions and solve various mathematical problems.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to find a common denominator to combine fractions.
  • Use factoring to simplify the numerator and denominator.
  • Combine like terms to simplify the expression further.
  • Use algebraic manipulations to solve mathematical problems.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

  • Physics: Simplifying algebraic expressions is crucial in solving problems related to motion, energy, and momentum.
  • Engineering: Algebraic manipulations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Simplifying algebraic expressions is essential in modeling economic systems and making predictions about economic trends.

Final Thoughts

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. By following the steps outlined in this article, you can simplify complex algebraic expressions and solve mathematical problems with ease. Remember to find a common denominator, use factoring, combine like terms, and use algebraic manipulations to simplify the expression further. With practice and patience, you will become proficient in simplifying algebraic expressions and solving mathematical problems.

Introduction

In our previous article, we explored the process of simplifying the expression: xx(x+2)−2(x−1)\frac{x}{x(x+2)} - \frac{2}{(x-1)}. We broke down the expression into manageable steps, and by the end of the article, you had a clear understanding of how to simplify complex algebraic expressions. In this article, we will address some of the most frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the least common multiple (LCM) of the denominators. This will allow you to find a common denominator and combine the fractions.

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

A: To find the LCM of two expressions, you can list the factors of each expression and identify the common factors. The LCM is the product of the highest power of each common factor.

Q: What is the difference between a common denominator and a least common multiple?

A: A common denominator is a denominator that is shared by two or more fractions, while a least common multiple (LCM) is the smallest multiple that is common to two or more numbers.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you can factor the denominator and cancel out any common factors between the numerator and the denominator.

Q: What is the purpose of combining like terms in an algebraic expression?

A: Combining like terms in an algebraic expression allows you to simplify the expression by eliminating any unnecessary terms.

Q: How do I know when to use algebraic manipulations to simplify an expression?

A: You should use algebraic manipulations to simplify an expression when the expression is complex and cannot be simplified using basic arithmetic operations.

Q: What are some common algebraic manipulations used to simplify expressions?

A: Some common algebraic manipulations used to simplify expressions include factoring, canceling, and combining like terms.

Q: How do I know if an expression is simplified?

A: An expression is simplified when it cannot be simplified further using basic arithmetic operations or algebraic manipulations.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications, including physics, engineering, and economics.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to find a common denominator to combine fractions.
  • Use factoring to simplify the numerator and denominator.
  • Combine like terms to simplify the expression further.
  • Use algebraic manipulations to solve mathematical problems.
  • Practice, practice, practice! The more you practice simplifying algebraic expressions, the more comfortable you will become with the process.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving various mathematical problems. By following the steps outlined in this article, you can simplify complex algebraic expressions and solve mathematical problems with ease. Remember to find a common denominator, use factoring, combine like terms, and use algebraic manipulations to simplify the expression further. With practice and patience, you will become proficient in simplifying algebraic expressions and solving mathematical problems.

Final Thoughts

Simplifying algebraic expressions is a skill that takes time and practice to develop. With patience and persistence, you can become proficient in simplifying complex algebraic expressions and solving mathematical problems. Remember to always follow the steps outlined in this article, and don't be afraid to ask for help when you need it.