Simplify The Following Expression:$\frac{3x}{x-3} - \frac{2x}{x^2-6x+9}$

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Introduction

Algebraic expressions can be complex and daunting, especially when dealing with fractions and variables. In this article, we will focus on simplifying a specific expression involving fractions and variables. We will break down the process into manageable steps, making it easier to understand and follow along.

The Expression to Simplify

The given expression is:

3xxβˆ’3βˆ’2xx2βˆ’6x+9\frac{3x}{x-3} - \frac{2x}{x^2-6x+9}

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of both fractions. The first denominator, xβˆ’3x-3, is already factored. The second denominator, x2βˆ’6x+9x^2-6x+9, can be factored as a perfect square trinomial:

x2βˆ’6x+9=(xβˆ’3)2x^2-6x+9 = (x-3)^2

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression:

3xxβˆ’3βˆ’2x(xβˆ’3)2\frac{3x}{x-3} - \frac{2x}{(x-3)^2}

Step 3: Find a Common Denominator

To combine the fractions, we need to find a common denominator. In this case, the common denominator is (xβˆ’3)2(x-3)^2. We can rewrite the first fraction with the common denominator:

3x(xβˆ’3)(xβˆ’3)2βˆ’2x(xβˆ’3)2\frac{3x(x-3)}{(x-3)^2} - \frac{2x}{(x-3)^2}

Step 4: Combine the Fractions

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

3x(xβˆ’3)βˆ’2x(xβˆ’3)2\frac{3x(x-3) - 2x}{(x-3)^2}

Step 5: Simplify the Numerator

We can simplify the numerator by distributing the 3x3x and combining like terms:

3x(xβˆ’3)βˆ’2x=3x2βˆ’9xβˆ’2x=3x2βˆ’11x3x(x-3) - 2x = 3x^2 - 9x - 2x = 3x^2 - 11x

Step 6: Rewrite the Simplified Expression

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

3x2βˆ’11x(xβˆ’3)2\frac{3x^2 - 11x}{(x-3)^2}

Conclusion

Simplifying complex algebraic expressions requires patience and attention to detail. By following the steps outlined in this article, we were able to simplify the given expression. Remember to factor the denominators, find a common denominator, combine the fractions, simplify the numerator, and rewrite the simplified expression.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS).
  • Factoring the denominators can help simplify the expression.
  • Finding a common denominator is crucial when combining fractions.
  • Simplifying the numerator can help reduce the complexity of the expression.

Real-World Applications

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

  • Physics: Simplifying expressions is essential in physics, where complex equations are used to describe the behavior of physical systems.
  • Engineering: Engineers use algebraic expressions to design and optimize systems, such as bridges and buildings.
  • Economics: Economists use algebraic expressions to model economic systems and make predictions about future trends.

Final Thoughts

Simplifying complex algebraic expressions is a valuable skill that can be applied to various fields. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to practice regularly and seek help when needed. With patience and persistence, you can master the art of simplifying algebraic expressions.

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Introduction

In our previous article, we explored the process of simplifying complex algebraic expressions. We broke down the steps involved in simplifying a specific expression and provided tips and tricks for success. In this article, we will answer some of the most 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 factor the denominators. This can help simplify the expression and make it easier to work with.

Q: How do I find a common denominator when simplifying fractions?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the two denominators. You can then rewrite each fraction with the common denominator.

Q: What is the difference between a perfect square trinomial and a non-perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored as the square of a binomial. For example, x2+6x+9x^2 + 6x + 9 can be factored as (x+3)2(x+3)^2. A non-perfect square trinomial, on the other hand, cannot be factored as the square of a binomial.

Q: How do I simplify a numerator when simplifying an algebraic expression?

A: To simplify a numerator, you need to combine like terms and eliminate any common factors. For example, if you have the expression 3x2+2x+4x3x^2 + 2x + 4x, you can combine the like terms to get 3x2+6x3x^2 + 6x.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it can help you:

  • Solve equations and inequalities more easily
  • Graph functions more accurately
  • Understand the behavior of functions and their derivatives
  • Make predictions about real-world phenomena

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working through practice problems in your textbook or online resources
  • Creating your own practice problems and solving them
  • Joining a study group or working with a tutor to get help and feedback

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to follow the order of operations (PEMDAS).
  • Factoring the denominators can help simplify the expression.
  • Finding a common denominator is crucial when combining fractions.
  • Simplifying the numerator can help reduce the complexity of the expression.

Real-World Applications

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

  • Physics: Simplifying expressions is essential in physics, where complex equations are used to describe the behavior of physical systems.
  • Engineering: Engineers use algebraic expressions to design and optimize systems, such as bridges and buildings.
  • Economics: Economists use algebraic expressions to model economic systems and make predictions about future trends.

Final Thoughts

Simplifying complex algebraic expressions is a valuable skill that can be applied to various fields. By following the steps outlined in this article and practicing regularly, you can master the art of simplifying algebraic expressions. Remember to seek help when needed and to be patient and persistent in your efforts.

Additional Resources

  • Khan Academy: Algebra
  • Mathway: Algebra Calculator
  • Wolfram Alpha: Algebra Solver

These resources can provide you with additional practice problems, tutorials, and tools to help you simplify algebraic expressions.