Simplifying Mathematical Expressions: A Guide to Simplifying 2/x - 2x/x^2 - 4 and y 3/x - 6

In the world of mathematics, simplifying expressions is an essential skill. Let's explore how to simplify complex expressions like 2/x - 2x/x^2 - 4 and how to put y 3/x - 6 in its simplest form. Understanding these concepts can enhance your problem-solving abilities and make complex equations more manageable.

1. Simplifying 2/x - 2x/x^2 - 4

One of the key steps in simplifying any expression is to find a common denominator. Let's break down the process step by step:

Step 1: Factor the expression:
2/x - 2x/x^2 - 4 Step 2: Multiply the top and bottom of the first term (2/x) by x^2:
2x^2 / x^2 - 2x/x^2 - 4 Step 3: Combine the terms:
(2x^2 - 2x) / (x^2 - 4) Step 4: Factor out the numerator:
(2x(x - 1)) / (x^2 - 4) Step 5: Simplify the expression:
3x^4 / (x^2 - 4)

The final simplified expression is 3x^4 / (x^2 - 4).

2. Simplifying y 3/x - 6

When dealing with expressions like y 3/x - 6, the goal is to put it into its simplest form. Here’s a straightforward way to approach this:

Step 1: Identify like terms:
y 2/x - 2x/x^2 - 4 Step 2: Factor and simplify:
2/x - 21/x - 4 2/x - 1/x - 2 - 4 2/x - 1/x - 6 Step 3: Combine terms:
y 3/x - 6

This expression is already in its simplest form, y 3/x - 6.

Conclusion

Simplifying complex expressions can greatly enhance your understanding of algebraic functions and make problem-solving more effective. By following these steps and understanding the underlying principles, you can tackle more advanced mathematical challenges with confidence. If you're looking for more resources or explanations, consider studying additional algebraic expressions and practicing with similar problems.