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

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: $\frac{1}{2x^2 - 4x} - \frac{2}{x}$. We will break down the solution into manageable steps, making it easy to understand and follow along.

Step 1: Factor the Denominator

The first step in simplifying the given expression is to factor the denominator of the first fraction. We can factor out a $2x$ from the denominator:

$$\frac{1}{2x^2 - 4x} = \frac{1}{2x(x - 2)}$$

Step 2: Find a Common Denominator

To subtract the two fractions, we need to find a common denominator. The common denominator is the product of the two denominators:

$$2x(x - 2) \cdot x = 2x^2(x - 2)$$

Step 3: Rewrite the Fractions with the Common Denominator

Now that we have a common denominator, we can rewrite the fractions:

$$\frac{1}{2x(x - 2)} - \frac{2}{x} = \frac{1 \cdot x}{2x(x - 2)} - \frac{2 \cdot 2x(x - 2)}{x \cdot 2x(x - 2)}$$

Step 4: Simplify the Expression

Now that we have the fractions with a common denominator, we can simplify the expression by combining the numerators:

$$\frac{x - 4x(x - 2)}{2x(x - 2)}$$

Step 5: Simplify the Numerator

We can simplify the numerator by combining like terms:

$$x - 4x(x - 2) = x - 4x^2 + 8x$$

$$= -4x^2 + 9x$$

Step 6: Write the Final Answer

Now that we have simplified the expression, we can write the final answer:

$$\frac{-4x^2 + 9x}{2x(x - 2)}$$

Conclusion

Simplifying algebraic expressions requires a step-by-step approach. By factoring the denominator, finding a common denominator, rewriting the fractions, simplifying the expression, and simplifying the numerator, we can arrive at the final answer. In this article, we simplified the expression $\frac{1}{2x^2 - 4x} - \frac{2}{x}$ and arrived at the final answer: $\frac{-4x^2 + 9x}{2x(x - 2)}$.

Answer Key

The correct answer is:

• B. $\frac{-4x^2 + 9x}{2x(x - 2)}$

Discussion

This problem requires a deep understanding of algebraic expressions and simplification techniques. The student should be able to factor the denominator, find a common denominator, rewrite the fractions, simplify the expression, and simplify the numerator. The student should also be able to identify the correct answer from the given options.

Tips and Variations

• To make this problem more challenging, you can add more complex expressions or require the student to simplify the expression using different techniques.
• To make this problem easier, you can provide additional hints or guidance throughout the solution.
• You can also use this problem as a starting point to explore other algebraic concepts, such as solving equations or graphing functions.

Real-World Applications

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

• Science and Engineering: Algebraic expressions are used to model real-world phenomena, such as the motion of objects or the behavior of electrical circuits.
• Economics: Algebraic expressions are used to model economic systems, such as supply and demand curves or investment portfolios.
• Computer Science: Algebraic expressions are used to develop algorithms and data structures, such as sorting and searching algorithms.

Introduction

Simplifying algebraic expressions is a fundamental concept in mathematics, and it's essential to understand the techniques and strategies involved. In this article, we'll provide a Q&A guide to help you master the art of simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It's a way to represent a mathematical relationship between variables and constants.

Q: What are the steps to simplify an algebraic expression?

A: The steps to simplify an algebraic expression are:

1. Factor the denominator: Factor out any common factors from the denominator.
2. Find a common denominator: Find a common denominator for the fractions.
3. Rewrite the fractions: Rewrite the fractions with the common denominator.
4. Simplify the expression: Simplify the expression by combining like terms.
5. Simplify the numerator: Simplify the numerator by combining like terms.

Q: How do I factor the denominator?

A: To factor the denominator, look for any common factors that can be factored out. For example, if the denominator is $2x^2 - 4x$, you can factor out a $2x$:

$$2x^2 - 4x = 2x(x - 2)$$

Q: How do I find a common denominator?

A: To find a common denominator, multiply the denominators of the fractions together. For example, if the fractions are $\frac{1}{2x}$ and $\frac{2}{x}$, the common denominator is $2x \cdot x = 2x^2$.

Q: How do I rewrite the fractions?

A: To rewrite the fractions, multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator. For example, if the fractions are $\frac{1}{2x}$ and $\frac{2}{x}$, you can rewrite them as:

$$\frac{1}{2x} = \frac{1 \cdot x}{2x \cdot x} = \frac{x}{2x^2}$$

$$\frac{2}{x} = \frac{2 \cdot 2x}{x \cdot 2x} = \frac{4x}{2x^2}$$

Q: How do I simplify the expression?

A: To simplify the expression, combine like terms in the numerator. For example, if the expression is $\frac{x}{2x^2} - \frac{4x}{2x^2}$, you can simplify it by combining the like terms:

$$\frac{x}{2x^2} - \frac{4x}{2x^2} = \frac{x - 4x}{2x^2} = \frac{-3x}{2x^2}$$

Q: How do I simplify the numerator?

A: To simplify the numerator, combine like terms. For example, if the numerator is $x - 4x$, you can simplify it by combining the like terms:

$$x - 4x = -3x$$

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

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

• Not factoring the denominator: Failing to factor the denominator can lead to incorrect simplifications.
• Not finding a common denominator: Failing to find a common denominator can lead to incorrect simplifications.
• Not rewriting the fractions: Failing to rewrite the fractions can lead to incorrect simplifications.
• Not simplifying the expression: Failing to simplify the expression can lead to incorrect simplifications.
• Not simplifying the numerator: Failing to simplify the numerator can lead to incorrect simplifications.

Conclusion

Simplifying algebraic expressions is a fundamental concept in mathematics, and it's essential to understand the techniques and strategies involved. By following the steps outlined in this Q&A guide, you can master the art of simplifying algebraic expressions and apply them to real-world problems.

Tips and Variations

• To make this problem more challenging, you can add more complex expressions or require the student to simplify the expression using different techniques.
• To make this problem easier, you can provide additional hints or guidance throughout the solution.
• You can also use this problem as a starting point to explore other algebraic concepts, such as solving equations or graphing functions.

Real-World Applications

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

• Science and Engineering: Algebraic expressions are used to model real-world phenomena, such as the motion of objects or the behavior of electrical circuits.
• Economics: Algebraic expressions are used to model economic systems, such as supply and demand curves or investment portfolios.
• Computer Science: Algebraic expressions are used to develop algorithms and data structures, such as sorting and searching algorithms.

By mastering the art of simplifying algebraic expressions, students can develop a deeper understanding of mathematical concepts and apply them to real-world problems.