Select The Correct Answer.What Is This Expression In Simplest Form? 1 2 X 2 − 4 X − 2 X \frac{1}{2x^2-4x} - \frac{2}{x} 2 X 2 − 4 X 1 ​ − X 2 ​ A. − 1 2 X ( X − 2 ) \frac{-1}{2x(x-2)} 2 X ( X − 2 ) − 1 ​ B. 4 X − 7 2 X ( X − 2 ) \frac{4x-7}{2x(x-2)} 2 X ( X − 2 ) 4 X − 7 ​ C. − 3 X − 8 2 X ( X − 2 ) \frac{-3x-8}{2x(x-2)} 2 X ( X − 2 ) − 3 X − 8 ​ D. − 4 X + 9 2 X ( X − 2 ) \frac{-4x+9}{2x(x-2)} 2 X ( X − 2 ) − 4 X + 9 ​

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression 12x2−4x−2x\frac{1}{2x^2-4x} - \frac{2}{x}. We will break down the steps involved in simplifying this expression and provide a clear explanation of the process.

Understanding the Expression

The given expression is 12x2−4x−2x\frac{1}{2x^2-4x} - \frac{2}{x}. To simplify this expression, we need to first understand the concept of a common denominator. A common denominator is the least common multiple (LCM) of the denominators of two or more fractions. In this case, the denominators are 2x2−4x2x^2-4x and xx.

Finding the Common Denominator

To find the common denominator, we need to factorize the denominators. The denominator 2x2−4x2x^2-4x can be factorized as 2x(x−2)2x(x-2). The denominator xx can be written as x⋅1x \cdot 1. Therefore, the common denominator is 2x(x−2)2x(x-2).

Rewriting the Expression

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

12x2−4x−2x=12x(x−2)−2x⋅xx\frac{1}{2x^2-4x} - \frac{2}{x} = \frac{1}{2x(x-2)} - \frac{2}{x} \cdot \frac{x}{x}

Simplifying the Expression

To simplify the expression, we need to combine the two fractions. We can do this by finding a common denominator and then adding or subtracting the numerators.

12x(x−2)−2x⋅xx=1⋅x−2⋅2x2x(x−2)\frac{1}{2x(x-2)} - \frac{2}{x} \cdot \frac{x}{x} = \frac{1 \cdot x - 2 \cdot 2x}{2x(x-2)}

Simplifying the Numerator

The numerator of the expression is 1⋅x−2⋅2x1 \cdot x - 2 \cdot 2x. We can simplify this by combining like terms.

1⋅x−2⋅2x=x−4x=−3x1 \cdot x - 2 \cdot 2x = x - 4x = -3x

Final Simplified Expression

Therefore, the final simplified expression is:

12x2−4x−2x=−3x2x(x−2)\frac{1}{2x^2-4x} - \frac{2}{x} = \frac{-3x}{2x(x-2)}

Simplifying the Expression Further

We can simplify the expression further by factoring out a negative sign from the numerator.

−3x2x(x−2)=−3x2x(x−2)⋅−1−1\frac{-3x}{2x(x-2)} = \frac{-3x}{2x(x-2)} \cdot \frac{-1}{-1}

Final Answer

Therefore, the final simplified expression is:

12x2−4x−2x=−3x2x(x−2)=−3x2x(x−2)⋅−1−1=−3x2x(x−2)\frac{1}{2x^2-4x} - \frac{2}{x} = \frac{-3x}{2x(x-2)} = \frac{-3x}{2x(x-2)} \cdot \frac{-1}{-1} = \frac{-3x}{2x(x-2)}

Conclusion

In this article, we have explored the process of simplifying algebraic expressions, with a focus on the given expression 12x2−4x−2x\frac{1}{2x^2-4x} - \frac{2}{x}. We have broken down the steps involved in simplifying this expression and provided a clear explanation of the process. The final simplified expression is −3x2x(x−2)\frac{-3x}{2x(x-2)}.

Answer Options

The answer options are:

A. −12x(x−2)\frac{-1}{2x(x-2)} B. 4x−72x(x−2)\frac{4x-7}{2x(x-2)} C. −3x−82x(x−2)\frac{-3x-8}{2x(x-2)} D. −4x+92x(x−2)\frac{-4x+9}{2x(x-2)}

Correct Answer

The correct answer is:

A. −12x(x−2)\frac{-1}{2x(x-2)}

However, we have simplified the expression to −3x2x(x−2)\frac{-3x}{2x(x-2)}. Therefore, the correct answer is:

−3x2x(x−2)\frac{-3x}{2x(x-2)}

Note

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression 12x2−4x−2x\frac{1}{2x^2-4x} - \frac{2}{x}. We broke down the steps involved in simplifying this expression and provided a clear explanation of the process. In this article, we will answer some frequently asked questions about 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 common denominator. The common denominator is the least common multiple (LCM) of the denominators of two or more fractions.

Q: How do I find the common denominator?

A: To find the common denominator, you need to factorize the denominators. Once you have factorized the denominators, you can find the LCM by multiplying the highest power of each factor.

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, you need to rewrite the expression with the common denominator. This involves multiplying the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

Q: How do I simplify the expression after rewriting it with the common denominator?

A: To simplify the expression, you need to combine the two fractions. You can do this by adding or subtracting the numerators and keeping the common denominator the same.

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

A: The final step in simplifying an algebraic expression is to simplify the numerator by combining like terms.

Q: Can I simplify an algebraic expression further?

A: Yes, you can simplify an algebraic expression further by factoring out a negative sign from the numerator or by canceling out any common factors between the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not identifying the common denominator
  • Not rewriting the expression with the common denominator
  • Not combining the fractions correctly
  • Not simplifying the numerator by combining like terms

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources or math software to help you practice and check your work.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions. Remember to identify the common denominator, rewrite the expression with the common denominator, combine the fractions, simplify the numerator, and check your work. With practice and patience, you can become proficient in simplifying algebraic expressions.

Additional Resources

  • Online resources: Khan Academy, Mathway, Wolfram Alpha
  • Math software: Mathematica, Maple, MATLAB
  • Textbooks: Algebra and Trigonometry by Michael Sullivan, College Algebra by James Stewart

Final Tips

  • Always check your work by plugging in values or using a calculator.
  • Practice simplifying algebraic expressions regularly to build your skills and confidence.
  • Don't be afraid to ask for help if you're struggling with a particular concept or problem.