Simplify The Expression:$\[ \frac{x-4}{3x^2-12x} \\]

Mar 1, 2025 by ADMIN 53 views

Simplify the Expression: A Step-by-Step Guide to Algebraic Manipulation

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will focus on simplifying the given expression: $\frac{x-4}{3x^2-12x}$ . We will break down the process into manageable steps, making it easier to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at it. The given expression is a fraction, where the numerator is $x-4$ and the denominator is $3x^2-12x$ . Our goal is to simplify this expression, making it easier to work with and understand.

Step 1: Factor the Denominator

To simplify the expression, we need to factor the denominator. The denominator is a quadratic expression, which can be factored as follows:

3x^2-12x = 3x(x-4)

By factoring the denominator, we have simplified the expression slightly, but we still have a long way to go.

Step 2: Cancel Common Factors

Now that we have factored the denominator, we can cancel common factors between the numerator and the denominator. In this case, we can cancel the common factor of $(x-4)$ :

\frac{x-4}{3x^2-12x} = \frac{x-4}{3x(x-4)}

By canceling the common factor of $(x-4)$ , we have simplified the expression further.

Step 3: Simplify the Expression

Now that we have canceled the common factor, we can simplify the expression further. We can cancel the common factor of $3x$ :

\frac{x-4}{3x(x-4)} = \frac{1}{3x}

By simplifying the expression, we have arrived at our final answer.

Conclusion

Simplifying the expression $\frac{x-4}{3x^2-12x}$ requires careful attention to detail and a solid understanding of algebraic manipulation. By factoring the denominator, canceling common factors, and simplifying the expression, we have arrived at our final answer: $\frac{1}{3x}$ . This expression is much simpler and easier to work with than the original expression.

Tips and Tricks

When simplifying expressions, always look for common factors between the numerator and the denominator.
Factoring the denominator can help simplify the expression.
Canceling common factors can help simplify the expression further.
Simplifying the expression can make it easier to work with and understand.

Real-World Applications

Simplifying expressions has many real-world applications, including:

Physics: Simplifying expressions is crucial in physics, where complex equations need to be solved to understand the behavior of physical systems.
Engineering: Simplifying expressions is essential in engineering, where complex systems need to be designed and analyzed.
Computer Science: Simplifying expressions is a fundamental concept in computer science, where algorithms need to be designed and analyzed.

Final Thoughts

Simplifying expressions is a crucial aspect of mathematics, and it requires careful attention to detail and a solid understanding of algebraic manipulation. By following the steps outlined in this article, you can simplify complex expressions and arrive at your final answer. Remember to always look for common factors, factor the denominator, cancel common factors, and simplify the expression to arrive at your final answer.

Frequently Asked Questions

Q: What is the final answer to the expression $\frac{x-4}{3x^2-12x}$ ? A: The final answer is $\frac{1}{3x}$ .
Q: How do I simplify the expression $\frac{x-4}{3x^2-12x}$ ? A: To simplify the expression, factor the denominator, cancel common factors, and simplify the expression.
Q: What are some real-world applications of simplifying expressions? A: Simplifying expressions has many real-world applications, including physics, engineering, and computer science.

References

[1] Algebraic Manipulation, Wikipedia
[2] Simplifying Expressions, Khan Academy
[3] Algebra, MIT OpenCourseWare

Additional Resources

[1] Algebraic Manipulation, Mathway
[2] Simplifying Expressions, Wolfram Alpha
[3] Algebra, Coursera

Introduction

In our previous article, we explored the process of simplifying the expression $\frac{x-4}{3x^2-12x}$ . We broke down the process into manageable steps, making it easier to understand and follow along. In this article, we will continue to explore the topic of simplifying expressions, this time in the form of a Q&A guide.

Q&A Guide

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

A: The first step in simplifying an expression is to factor the denominator. This can help simplify the expression and make it easier to work with.

Q: How do I factor the denominator?

A: To factor the denominator, look for common factors between the numerator and the denominator. If you find a common factor, you can cancel it out to simplify the expression.

Q: What is the next step in simplifying an expression?

A: After factoring the denominator, the next step is to cancel common factors between the numerator and the denominator. This can help simplify the expression further.

Q: How do I cancel common factors?

A: To cancel common factors, look for common factors between the numerator and the denominator. If you find a common factor, you can cancel it out to simplify the expression.

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

A: The final step in simplifying an expression is to simplify the expression itself. This can involve canceling out any remaining common factors or simplifying the expression further.

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

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

Not factoring the denominator
Not canceling common factors
Not simplifying the expression itself
Making errors when canceling common factors

Q: How do I know if I have simplified an expression correctly?

A: To know if you have simplified an expression correctly, check your work by plugging the simplified expression back into the original equation. If the simplified expression is true, then you have simplified the expression correctly.

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

A: Simplifying expressions has many real-world applications, including physics, engineering, and computer science.

Q: How do I practice simplifying expressions?

A: To practice simplifying expressions, try working through examples and exercises. You can also use online resources, such as math websites and apps, to practice simplifying expressions.