Add The Following Expressions:$\[ \frac{4x+7}{x^2+2} + \frac{3x-4}{x^2+2} \\]Choose The Correct Result:A. \[$\frac{11x-1}{x^2+2}\$\]B. \[$\frac{7x-11}{x^2+2}\$\]C. \[$\frac{7x+3}{x^2+2}\$\]D.

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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 how to simplify algebraic expressions, focusing on the given problem: 4x+7x2+2+3xβˆ’4x2+2\frac{4x+7}{x^2+2} + \frac{3x-4}{x^2+2}. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding the Problem

The given problem involves adding two fractions with the same denominator, x2+2x^2+2. To simplify the expression, we need to combine the numerators while keeping the denominator the same.

Step 1: Identify the Numerators

The first step is to identify the numerators of the two fractions. The numerators are the expressions on top of the fractions.

  • The numerator of the first fraction is 4x+74x+7.
  • The numerator of the second fraction is 3xβˆ’43x-4.

Step 2: Combine the Numerators

Now that we have identified the numerators, we can combine them by adding the two expressions.

4x+7x2+2+3xβˆ’4x2+2=(4x+7)+(3xβˆ’4)x2+2\frac{4x+7}{x^2+2} + \frac{3x-4}{x^2+2} = \frac{(4x+7) + (3x-4)}{x^2+2}

Step 3: Simplify the Numerator

To simplify the numerator, we need to combine like terms.

(4x+7)+(3xβˆ’4)=7x+3(4x+7) + (3x-4) = 7x + 3

So, the simplified expression becomes:

7x+3x2+2\frac{7x+3}{x^2+2}

Conclusion

In conclusion, the correct result for the given problem is 7x+3x2+2\frac{7x+3}{x^2+2}. This expression is the simplified form of the original problem, and it is the correct answer among the options provided.

Discussion

The given problem is a classic example of simplifying algebraic expressions. By following the steps outlined above, we can simplify even the most complex expressions. The key is to identify the numerators, combine them, and simplify the resulting expression.

Common Mistakes

When simplifying algebraic expressions, it's essential to avoid common mistakes. Some common mistakes include:

  • Failing to combine like terms
  • Not simplifying the numerator
  • Not keeping the denominator the same

By avoiding these mistakes, we can ensure that our solutions are accurate and complete.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. In physics, for example, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. By mastering the art of simplifying algebraic expressions, we can solve complex problems and make informed decisions.

Final Thoughts

In conclusion, simplifying algebraic expressions is a crucial skill for any math enthusiast. By following the steps outlined above, we can simplify even the most complex expressions. Remember to identify the numerators, combine them, and simplify the resulting expression. With practice and patience, you'll become a master of simplifying algebraic expressions in no time.

Options Analysis

Let's analyze the options provided:

  • Option A: 11xβˆ’1x2+2\frac{11x-1}{x^2+2}
  • Option B: 7xβˆ’11x2+2\frac{7x-11}{x^2+2}
  • Option C: 7x+3x2+2\frac{7x+3}{x^2+2}
  • Option D: (Not provided)

Based on our solution, we can see that the correct answer is Option C: 7x+3x2+2\frac{7x+3}{x^2+2}.

Conclusion

Introduction

In our previous article, we explored how to simplify algebraic expressions, focusing on the given problem: 4x+7x2+2+3xβˆ’4x2+2\frac{4x+7}{x^2+2} + \frac{3x-4}{x^2+2}. We broke down the solution step by step, using clear and concise language to ensure that readers understand the process. In this article, we will continue to provide a Q&A guide on simplifying algebraic expressions.

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

A: The first step in simplifying an algebraic expression is to identify the numerators and denominators of the fractions involved.

Q: How do I combine the numerators of two fractions with the same denominator?

A: To combine the numerators of two fractions with the same denominator, you need to add the two expressions. For example, if you have ab+cb\frac{a}{b} + \frac{c}{b}, you can combine the numerators by adding aa and cc.

Q: What is the next step after combining the numerators?

A: After combining the numerators, you need to simplify the resulting expression by combining like terms.

Q: How do I simplify an algebraic expression with multiple terms?

A: To simplify an algebraic expression with multiple terms, you need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, 2x2x and 3x3x are like terms, but 2x2x and 3y3y are not.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to solve
  • Avoid errors
  • Improve understanding of the underlying math concepts

Q: Can I simplify an algebraic expression with a negative exponent?

A: Yes, you can simplify an algebraic expression with a negative exponent. To do this, you need to rewrite the expression with a positive exponent and then simplify.

Q: How do I simplify an algebraic expression with a fraction in the denominator?

A: To simplify an algebraic expression with a fraction in the denominator, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator.

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

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

  • Failing to combine like terms
  • Not simplifying the numerator
  • Not keeping the denominator the same
  • Making errors when multiplying or dividing fractions

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working through examples and exercises
  • Using online resources and tools
  • Practicing with real-world problems
  • Joining a study group or seeking help from a tutor

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined above and practicing regularly, you can become proficient in simplifying algebraic expressions and apply this skill to a wide range of problems.

Additional Resources

  • Algebraic Expression Simplification Worksheet: A worksheet with examples and exercises to help you practice simplifying algebraic expressions.
  • Algebraic Expression Simplification Video Tutorial: A video tutorial that provides a step-by-step guide on simplifying algebraic expressions.
  • Algebraic Expression Simplification Online Tool: An online tool that allows you to enter an algebraic expression and simplify it.

Final Thoughts

In conclusion, simplifying algebraic expressions is a crucial skill for any math enthusiast. By mastering this skill, you can solve complex problems and make informed decisions in various fields. Remember to practice regularly and seek help when needed to become proficient in simplifying algebraic expressions.