Solve The Equation For { N $} : : : { \frac{1}{n^2} + \frac{1}{n} = \frac{1}{2n^2} \}

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Introduction

In this article, we will delve into solving a quadratic equation involving fractions. The given equation is 1n2+1n=12n2\frac{1}{n^2} + \frac{1}{n} = \frac{1}{2n^2}. Our goal is to find the value of nn that satisfies this equation. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Multiply Both Sides by the Least Common Denominator

To simplify the equation, we need to eliminate the fractions. The least common denominator (LCD) of the fractions is 2n22n^2. We will multiply both sides of the equation by 2n22n^2 to get rid of the fractions.

2n2â‹…(1n2+1n)=2n2â‹…12n22n^2 \cdot \left(\frac{1}{n^2} + \frac{1}{n}\right) = 2n^2 \cdot \frac{1}{2n^2}

This simplifies to:

2+2n=12 + 2n = 1

Step 2: Subtract 2 from Both Sides

Next, we will subtract 2 from both sides of the equation to isolate the term with nn.

2n=1−22n = 1 - 2

This simplifies to:

2n=−12n = -1

Step 3: Divide Both Sides by 2

Now, we will divide both sides of the equation by 2 to solve for nn.

n=−12n = \frac{-1}{2}

Conclusion

We have successfully solved the equation for nn. The value of nn that satisfies the equation is n=−12n = \frac{-1}{2}. This is the solution to the given equation.

Understanding the Solution

To understand the solution, let's analyze the original equation. The equation involves fractions, which can make it difficult to solve. By multiplying both sides by the LCD, we were able to eliminate the fractions and simplify the equation. This made it easier to solve for nn.

Real-World Applications

Solving equations like this one has real-world applications in various fields, such as physics, engineering, and economics. For example, in physics, equations like this one can be used to model the motion of objects. In engineering, equations like this one can be used to design and optimize systems. In economics, equations like this one can be used to model the behavior of markets.

Tips and Tricks

When solving equations like this one, it's essential to follow the order of operations (PEMDAS). This means that we need to perform the operations in the correct order, which is:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

By following the order of operations, we can ensure that we get the correct solution.

Common Mistakes

When solving equations like this one, there are several common mistakes that we can make. These include:

  • Not following the order of operations
  • Not simplifying the equation enough
  • Not checking the solution

To avoid these mistakes, it's essential to be careful and methodical when solving equations.

Conclusion

Introduction

In our previous article, we solved the equation 1n2+1n=12n2\frac{1}{n^2} + \frac{1}{n} = \frac{1}{2n^2} for nn. We broke down the solution into manageable steps and provided a step-by-step guide to solving the equation. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the least common denominator (LCD) of the fractions in the equation?

A: The LCD of the fractions in the equation is 2n22n^2. This is the smallest multiple of the denominators that can be divided evenly by each of the denominators.

Q: Why do we need to multiply both sides of the equation by the LCD?

A: We need to multiply both sides of the equation by the LCD to eliminate the fractions. This makes it easier to solve for nn.

Q: What is the next step after multiplying both sides by the LCD?

A: After multiplying both sides by the LCD, we need to subtract 2 from both sides of the equation to isolate the term with nn.

Q: Why do we need to divide both sides of the equation by 2?

A: We need to divide both sides of the equation by 2 to solve for nn. This is the final step in solving the equation.

Q: What is the value of nn that satisfies the equation?

A: The value of nn that satisfies the equation is n=−12n = \frac{-1}{2}.

Q: How can I apply this solution to real-world problems?

A: This solution can be applied to various real-world problems, such as modeling the motion of objects in physics, designing and optimizing systems in engineering, and modeling the behavior of markets in economics.

Q: What are some common mistakes to avoid when solving equations like this one?

A: Some common mistakes to avoid when solving equations like this one include not following the order of operations, not simplifying the equation enough, and not checking the solution.

Q: How can I ensure that I get the correct solution?

A: To ensure that you get the correct solution, it's essential to be careful and methodical when solving equations. Follow the order of operations, simplify the equation enough, and check the solution.

Q: Can I use this solution to solve other equations?

A: Yes, you can use this solution as a template to solve other equations. Just replace the values in the equation with the values from the new equation and follow the same steps.

Conclusion

Solving the equation for nn involves several steps, including multiplying both sides by the LCD, subtracting 2 from both sides, and dividing both sides by 2. By following these steps and being careful and methodical, you can ensure that you get the correct solution. This solution has real-world applications in various fields and can be used to model the behavior of objects, systems, and markets.

Additional Resources

If you have any further questions or need additional help, please refer to the following resources:

Final Thoughts

Solving equations like this one requires patience, persistence, and practice. By following the steps outlined in this article and being careful and methodical, you can ensure that you get the correct solution. Remember to apply this solution to real-world problems and to avoid common mistakes. Good luck!