Solve For { N $}$ In The Equation:${ -8 \sqrt{n+9} = -80 }$

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Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will focus on solving the equation ${-8 \sqrt{n+9} = -80}$ for the variable ${ n }$. This equation involves a square root, and our goal is to isolate the variable ${ n }$ and find its value.

Understanding the Equation

The given equation is ${-8 \sqrt{n+9} = -80}$. To start solving this equation, we need to understand the properties of square roots and how they interact with coefficients and variables. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have a square root of ${ n+9 }$, which means we are looking for a value that, when multiplied by itself, gives ${ n+9 }$.

Isolating the Square Root

To isolate the square root, we need to get rid of the coefficient ${-8}$ that is being multiplied by the square root. We can do this by dividing both sides of the equation by ${-8}$. This will give us:

${\sqrt{n+9} = 10}$

Squaring Both Sides

Now that we have isolated the square root, we can square both sides of the equation to get rid of the square root. Squaring both sides means multiplying both sides by themselves. This will give us:

${(\sqrt{n+9})^2 = 10^2}$

Simplifying the left-hand side, we get:

${n+9 = 100}$

Solving for ${ n }$

Now that we have simplified the equation, we can solve for ${ n }$ by subtracting ${9}$ from both sides of the equation. This will give us:

${n = 100 - 9}$

${n = 91}$

Conclusion

In this article, we solved the equation ${-8 \sqrt{n+9} = -80}$ for the variable ${ n }$. We started by isolating the square root, then squared both sides of the equation to get rid of the square root. Finally, we solved for ${ n }$ by subtracting ${9}$ from both sides of the equation. The value of ${ n }$ is ${91}$.

Tips and Tricks

• When solving equations involving square roots, it's essential to isolate the square root first.
• Squaring both sides of the equation can help get rid of the square root, but be careful not to introduce extraneous solutions.
• Always check your solutions by plugging them back into the original equation to ensure they are valid.

Common Mistakes

• Failing to isolate the square root before squaring both sides of the equation.
• Introducing extraneous solutions by squaring both sides of the equation without checking the solutions.
• Not checking the solutions by plugging them back into the original equation.

Real-World Applications

Solving equations involving square roots has many real-world applications, such as:

• Calculating distances and heights in geometry and trigonometry.
• Finding the area and perimeter of shapes in geometry.
• Solving problems involving rates and ratios in physics and engineering.

Final Thoughts

Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. By isolating the square root, squaring both sides of the equation, and solving for the variable, we can find the value of the variable. Remember to always check your solutions by plugging them back into the original equation to ensure they are valid.

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will answer some common questions related to solving equations involving square roots.

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

A: The first step in solving an equation involving a square root is to isolate the square root. This means getting the square root term by itself on one side of the equation.

Q: How do I isolate the square root?

A: To isolate the square root, you can use inverse operations to get rid of any coefficients or constants that are being multiplied by the square root. For example, if the equation is ${-8 \sqrt{n+9} = -80}$, you can divide both sides of the equation by ${-8}$ to get ${\sqrt{n+9} = 10}$.

Q: What happens when I square both sides of the equation?

A: When you square both sides of the equation, you are essentially multiplying both sides by themselves. This can help get rid of the square root, but be careful not to introduce extraneous solutions.

Q: How do I know if I have introduced an extraneous solution?

A: To check if you have introduced an extraneous solution, plug the solution back into the original equation and see if it is true. If the solution does not satisfy the original equation, then it is an extraneous solution.

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:

• Failing to isolate the square root before squaring both sides of the equation.
• Introducing extraneous solutions by squaring both sides of the equation without checking the solutions.
• Not checking the solutions by plugging them back into the original equation.

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

A: Solving equations involving square roots has many real-world applications, such as:

• Calculating distances and heights in geometry and trigonometry.
• Finding the area and perimeter of shapes in geometry.
• Solving problems involving rates and ratios in physics and engineering.

Q: Can you provide an example of how to solve an equation involving a square root?

A: Let's consider the equation ${\sqrt{x+16} = 5}$. To solve for ${x}$, we can square both sides of the equation to get:

${(\sqrt{x+16})^2 = 5^2}$

Simplifying the left-hand side, we get:

${x+16 = 25}$

Subtracting 16 from both sides, we get:

${x = 9}$

Q: How do I know if the solution is valid?

A: To check if the solution is valid, plug it back into the original equation and see if it is true. In this case, we can plug ${x = 9}$ back into the original equation to get:

${\sqrt{9+16} = 5}$

Simplifying the left-hand side, we get:

${\sqrt{25} = 5}$

Since this is true, the solution ${x = 9}$ is valid.

Q: What if I get a negative solution?

A: If you get a negative solution, it may not be valid. This is because the square root of a negative number is not a real number. In this case, you may need to re-examine your work and check for any errors.

Q: Can you provide some tips for solving equations involving square roots?

A: Here are some tips for solving equations involving square roots:

• Always isolate the square root before squaring both sides of the equation.
• Check your solutions by plugging them back into the original equation.
• Be careful not to introduce extraneous solutions.
• Use inverse operations to get rid of any coefficients or constants that are being multiplied by the square root.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. By isolating the square root, squaring both sides of the equation, and checking the solutions, we can find the value of the variable. Remember to always check your solutions by plugging them back into the original equation to ensure they are valid.