Solve N − 3 = 6 \sqrt{n-3}=6 N − 3 = 6 .Show Your Steps To Solve For N N N .

Mar 7, 2025 by ADMIN 79 views

**Solving Equations with Square Roots: A Step-by-Step Guide**

Introduction

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving the equation $\sqrt{n-3}=6$ and provide a step-by-step guide on how to find the value of $n$ .

Understanding the Equation

The given equation is $\sqrt{n-3}=6$ . This equation involves a square root, which means that the expression inside the square root must be non-negative. In other words, $n-3 \geq 0$ , which implies that $n \geq 3$ .

Step 1: Square Both Sides

To solve the equation, we need to get rid of the square root. We can do this by squaring both sides of the equation. This will eliminate the square root and allow us to solve for $n$ .

$\sqrt{n-3}=6$

Squaring both sides gives us:

$n-3=6^2$

$n-3=36$

Step 2: Add 3 to Both Sides

Now that we have $n-3=36$ , we need to isolate $n$ . We can do this by adding 3 to both sides of the equation.

$n-3+3=36+3$

$n=39$

Step 3: Check the Solution

Before we can be sure that our solution is correct, we need to check it. We can do this by plugging the value of $n$ back into the original equation.

$\sqrt{n-3}=6$

$\sqrt{39-3}=6$

$\sqrt{36}=6$

$6=6$

Since the left-hand side and right-hand side of the equation are equal, we can be sure that our solution is correct.

Conclusion

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article, we can solve the equation $\sqrt{n-3}=6$ and find the value of $n$ . Remember to always check your solution to ensure that it is correct.

Additional Tips and Tricks

When solving equations with square roots, make sure that the expression inside the square root is non-negative.
Use the property of square roots that $\sqrt{x^2}=x$ to simplify the equation.
Check your solution by plugging the value of $n$ back into the original equation.

Common Mistakes to Avoid

Don't forget to check your solution to ensure that it is correct.
Make sure that the expression inside the square root is non-negative.
Don't square both sides of the equation without checking that the expression inside the square root is non-negative.

Real-World Applications

Solving equations with square roots has many real-world applications. For example, in physics, the equation $\sqrt{v^2}=v$ is used to calculate the velocity of an object. In engineering, the equation $\sqrt{F^2}=F$ is used to calculate the force exerted on an object.

Conclusion

Final Answer

Introduction

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a Q&A guide on solving equations with square roots, including common mistakes to avoid and real-world applications.

Q: What is the first step in solving an equation with a square root?

A: The first step in solving an equation with a square root is to square both sides of the equation. This will eliminate the square root and allow us to solve for the variable.

Q: Why do I need to check my solution when solving an equation with a square root?

A: You need to check your solution when solving an equation with a square root because squaring both sides of the equation can introduce extraneous solutions. By plugging the value of the variable back into the original equation, you can ensure that your solution is correct.

Q: What is the property of square roots that I can use to simplify the equation?

A: The property of square roots that you can use to simplify the equation is $\sqrt{x^2}=x$ . This means that if the expression inside the square root is a perfect square, you can simplify the equation by taking the square root of both sides.

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

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

Forgetting to check your solution
Squaring both sides of the equation without checking that the expression inside the square root is non-negative
Not using the property of square roots to simplify the equation

Q: How do I know if the expression inside the square root is non-negative?

A: You can determine if the expression inside the square root is non-negative by checking if the value of the variable is greater than or equal to the constant term. For example, in the equation $\sqrt{x-4}=3$ , the expression inside the square root is non-negative if $x \geq 4$ .

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

A: Some real-world applications of solving equations with square roots include:

Calculating the velocity of an object in physics
Calculating the force exerted on an object in engineering
Determining the length of a shadow in geometry

Q: Can I use the property of square roots to solve equations with negative numbers?

A: Yes, you can use the property of square roots to solve equations with negative numbers. However, you need to be careful when taking the square root of a negative number, as it can introduce complex solutions.

Q: How do I handle complex solutions when solving equations with square roots?

A: When handling complex solutions, you need to remember that the square root of a negative number is an imaginary number. You can use the property of complex numbers to simplify the equation and find the solution.

Conclusion

Solving equations with square roots can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article and avoiding common mistakes, you can solve equations with square roots and find the value of the variable. Remember to always check your solution to ensure that it is correct.

Final Answer

The final answer is: $\boxed{39}$