Solve The Equation For { N $} : : : { \sqrt{2n - 3} = \sqrt{3n - 9} \}

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Introduction

In this article, we will delve into solving a quadratic equation involving square roots. The given equation is 2n−3=3n−9\sqrt{2n - 3} = \sqrt{3n - 9}, and our goal is to find the value of nn that satisfies this equation. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Square Both Sides of the Equation

The first step in solving this equation is to eliminate the square roots by squaring both sides. This will allow us to work with a simpler equation that is easier to manipulate.

(2n−3)2=(3n−9)2\left(\sqrt{2n - 3}\right)^2 = \left(\sqrt{3n - 9}\right)^2

Using the property of exponents that states (am)n=amn(a^m)^n = a^{mn}, we can simplify the equation to:

2n−3=3n−92n - 3 = 3n - 9

Step 2: Isolate the Variable n

Now that we have a linear equation, we can isolate the variable nn by moving all the terms involving nn to one side of the equation.

2n−3=3n−92n - 3 = 3n - 9

Subtracting 2n2n from both sides gives us:

−3=n−9-3 = n - 9

Step 3: Add 9 to Both Sides of the Equation

To isolate nn, we need to get rid of the constant term on the right-hand side. We can do this by adding 9 to both sides of the equation.

−3+9=n−9+9-3 + 9 = n - 9 + 9

Simplifying the equation gives us:

6=n6 = n

Step 4: Check the Solution

Before we conclude that n=6n = 6 is the solution to the equation, we need to check our work. We can do this by plugging the value of nn back into the original equation and verifying that it is true.

2(6)−3=3(6)−9\sqrt{2(6) - 3} = \sqrt{3(6) - 9}

Simplifying the equation gives us:

12−3=18−9\sqrt{12 - 3} = \sqrt{18 - 9}

9=9\sqrt{9} = \sqrt{9}

3=33 = 3

Since the equation is true, we can conclude that n=6n = 6 is indeed the solution to the equation.

Conclusion

In this article, we solved the equation 2n−3=3n−9\sqrt{2n - 3} = \sqrt{3n - 9} by squaring both sides, isolating the variable nn, and checking our work. We found that the value of nn that satisfies the equation is n=6n = 6. This solution can be verified by plugging it back into the original equation.

Additional Tips and Tricks

When solving equations involving square roots, it's essential to remember the following tips and tricks:

  • Square both sides: Squaring both sides of the equation is a powerful technique for eliminating square roots.
  • Isolate the variable: Make sure to isolate the variable on one side of the equation to avoid confusion.
  • Check your work: Always check your solution by plugging it back into the original equation to ensure that it is true.

By following these tips and tricks, you'll be well on your way to becoming a master of solving equations involving square roots.

Frequently Asked Questions

Here are some frequently asked questions about solving equations involving square roots:

  • Q: How do I know when to square both sides of the equation? A: You should square both sides of the equation when you have a square root on both sides and you want to eliminate the square roots.
  • Q: What if I get a negative value when I square both sides of the equation? A: If you get a negative value when you square both sides of the equation, it means that the original equation has no solution.
  • Q: How do I check my work when solving an equation involving square roots? A: To check your work, plug the value of the variable back into the original equation and verify that it is true.

By following these tips and tricks, you'll be able to solve equations involving square roots with confidence.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with practice and patience, you'll become a pro in no time. Remember to square both sides of the equation, isolate the variable, and check your work to ensure that your solution is correct. With these tips and tricks, you'll be well on your way to mastering the art of solving equations involving square roots.

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Introduction

In our previous article, we explored the step-by-step process of solving equations involving square roots. However, we understand that sometimes, it's not just about following a set of instructions, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we'll delve into a Q&A guide to help you better grasp the world of solving equations involving square roots.

Q: What is the main difference between solving equations involving square roots and solving linear equations?

A: The main difference between solving equations involving square roots and solving linear equations is the presence of square roots. When solving linear equations, you can simply add, subtract, multiply, or divide both sides of the equation to isolate the variable. However, when dealing with square roots, you need to square both sides of the equation to eliminate the square roots.

Q: How do I know when to square both sides of the equation?

A: You should square both sides of the equation when you have a square root on both sides and you want to eliminate the square roots. This is a crucial step in solving equations involving square roots, as it allows you to work with a simpler equation that is easier to manipulate.

Q: What if I get a negative value when I square both sides of the equation?

A: If you get a negative value when you square both sides of the equation, it means that the original equation has no solution. This is because the square of a real number is always non-negative, so if you get a negative value, it's a sign that the equation has no real solution.

Q: How do I check my work when solving an equation involving square roots?

A: To check your work, plug the value of the variable back into the original equation and verify that it is true. This is an essential step in ensuring that your solution is correct and that you haven't made any mistakes along the way.

Q: What are some common mistakes to avoid when solving equations involving square roots?

A: Some common mistakes to avoid when solving equations involving square roots include:

  • Squaring both sides of the equation without checking if the original equation has a real solution.
  • Failing to check your work by plugging the value of the variable back into the original equation.
  • Not considering the possibility of complex solutions when dealing with equations involving square roots.

Q: Can I use a calculator to solve equations involving square roots?

A: Yes, you can use a calculator to solve equations involving square roots. However, it's essential to understand the underlying concepts and to be able to verify your work using algebraic methods.

Q: How do I know if an equation involving square roots has a real solution?

A: To determine if an equation involving square roots has a real solution, you need to check if the expression inside the square root is non-negative. If it is, then the equation has a real solution. If it's not, then the equation has no real solution.

Q: Can I use the quadratic formula to solve equations involving square roots?

A: Yes, you can use the quadratic formula to solve equations involving square roots. However, you need to be careful when applying the quadratic formula, as it may not always yield a real solution.

Q: What are some real-world applications of solving equations involving square roots?

A: Solving equations involving square roots has numerous real-world applications, including:

  • Physics: When dealing with problems involving distance, speed, and time, you may need to solve equations involving square roots.
  • Engineering: In engineering, you may need to solve equations involving square roots when designing structures or systems that involve complex geometric shapes.
  • Computer Science: In computer science, you may need to solve equations involving square roots when working with algorithms that involve complex mathematical operations.

Conclusion

Solving equations involving square roots can be a challenging task, but with practice and patience, you'll become a pro in no time. Remember to square both sides of the equation, isolate the variable, and check your work to ensure that your solution is correct. By following these tips and tricks, you'll be well on your way to mastering the art of solving equations involving square roots.

Additional Resources

If you're looking for additional resources to help you learn more about solving equations involving square roots, here are some suggestions:

  • Online tutorials: Websites like Khan Academy, Coursera, and edX offer a wide range of online tutorials and courses on solving equations involving square roots.
  • Textbooks: There are many excellent textbooks available on solving equations involving square roots, including "Algebra" by Michael Artin and "Calculus" by Michael Spivak.
  • Practice problems: Websites like Mathway and Wolfram Alpha offer a wide range of practice problems on solving equations involving square roots.

By following these resources and practicing regularly, you'll be well on your way to becoming a master of solving equations involving square roots.