Solve For \[$ N \$\].$\[ \sqrt{10n + 1} = \sqrt{2n + 1} \\]

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Problem Statement

Given the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$, we need to solve for the variable $n$. This equation involves square roots, and our goal is to isolate the variable $n$ and find its value.

Understanding the Equation

The given equation is $\sqrt{10n + 1} = \sqrt{2n + 1}$. To solve for $n$, we need to eliminate the square roots. One way to do this is to square both sides of the equation. However, before we do that, let's understand the properties of square roots.

Properties of Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because $4 \times 4 = 16$. Similarly, the square root of 25 is 5, because $5 \times 5 = 25$.

Squaring Both Sides

Now that we understand the properties of square roots, let's square both sides of the equation. This will help us eliminate the square roots and solve for $n$.

$\left(\sqrt{10n + 1}\right)^2 = \left(\sqrt{2n + 1}\right)^2$

Using the property of square roots, we can simplify this equation as follows:

$10n + 1 = 2n + 1$

Simplifying the Equation

Now that we have squared both sides of the equation, let's simplify it further. We can start by subtracting $2n$ from both sides of the equation.

$10n - 2n + 1 = 2n - 2n + 1$

This simplifies to:

$8n + 1 = 1$

Solving for $n$

Now that we have simplified the equation, let's solve for $n$. We can start by subtracting 1 from both sides of the equation.

$8n = 0$

Next, we can divide both sides of the equation by 8.

$n = 0$

Conclusion

In this article, we solved the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$ for the variable $n$. We started by squaring both sides of the equation, which helped us eliminate the square roots. We then simplified the equation and solved for $n$. The final solution is $n = 0$.

Example Use Case

The equation $\sqrt{10n + 1} = \sqrt{2n + 1}$ can be used to model real-world problems. For example, suppose we are given a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We can use the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$ to find the value of $n$ that satisfies the quadratic equation.

Step-by-Step Solution

Here is a step-by-step solution to the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$:

1. Square both sides of the equation: $\left(\sqrt{10n + 1}\right)^2 = \left(\sqrt{2n + 1}\right)^2$
2. Simplify the equation: $10n + 1 = 2n + 1$
3. Subtract $2n$ from both sides of the equation: $10n - 2n + 1 = 2n - 2n + 1$
4. Simplify the equation: $8n + 1 = 1$
5. Subtract 1 from both sides of the equation: $8n = 0$
6. Divide both sides of the equation by 8: $n = 0$

Frequently Asked Questions

Here are some frequently asked questions about the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$:

• What is the value of $n$ that satisfies the equation? The value of $n$ that satisfies the equation is $n = 0$.
• How do I solve the equation? To solve the equation, you can square both sides of the equation, simplify it, and then solve for $n$.
• What is the significance of the equation? The equation can be used to model real-world problems, such as quadratic equations.

Conclusion

In this article, we solved the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$ for the variable $n$. We started by squaring both sides of the equation, which helped us eliminate the square roots. We then simplified the equation and solved for $n$. The final solution is $n = 0$.

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Frequently Asked Questions

In the previous article, we solved the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$ for the variable $n$. However, we understand that some readers may still have questions about the equation and its solution. In this article, we will address some of the most frequently asked questions about the equation.

Q: What is the value of $n$ that satisfies the equation?

A: The value of $n$ that satisfies the equation is $n = 0$.

Q: How do I solve the equation?

A: To solve the equation, you can square both sides of the equation, simplify it, and then solve for $n$. Here is a step-by-step solution:

1. Square both sides of the equation: $\left(\sqrt{10n + 1}\right)^2 = \left(\sqrt{2n + 1}\right)^2$
2. Simplify the equation: $10n + 1 = 2n + 1$
3. Subtract $2n$ from both sides of the equation: $10n - 2n + 1 = 2n - 2n + 1$
4. Simplify the equation: $8n + 1 = 1$
5. Subtract 1 from both sides of the equation: $8n = 0$
6. Divide both sides of the equation by 8: $n = 0$

Q: What is the significance of the equation?

A: The equation can be used to model real-world problems, such as quadratic equations.

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

A: Yes, you can use this equation as a starting point to solve other equations that involve square roots. However, you will need to modify the equation to fit the specific problem you are trying to solve.

Q: What if I get a different solution for $n$?

A: If you get a different solution for $n$, it may be because you made a mistake in your calculations or because the equation has multiple solutions. In this case, you will need to recheck your work and make sure that you are using the correct solution.

Q: Can I use this equation to solve equations with negative numbers?

A: Yes, you can use this equation to solve equations with negative numbers. However, you will need to be careful when working with negative numbers, as they can sometimes lead to extraneous solutions.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, you can plug it back into the original equation and make sure that it satisfies the equation. If it does, then your solution is correct. If it doesn't, then you will need to recheck your work and try again.

Q: Can I use this equation to solve equations with fractions?

A: Yes, you can use this equation to solve equations with fractions. However, you will need to be careful when working with fractions, as they can sometimes lead to extraneous solutions.

Q: How do I simplify the equation?

A: To simplify the equation, you can start by squaring both sides of the equation. This will help you eliminate the square roots and make it easier to solve for $n$.

Q: Can I use this equation to solve equations with decimals?

A: Yes, you can use this equation to solve equations with decimals. However, you will need to be careful when working with decimals, as they can sometimes lead to extraneous solutions.

Q: How do I know if my equation is quadratic?

A: To check if your equation is quadratic, you can look for the presence of a squared variable. If you see a squared variable, then your equation is quadratic.

Q: Can I use this equation to solve equations with absolute values?

A: Yes, you can use this equation to solve equations with absolute values. However, you will need to be careful when working with absolute values, as they can sometimes lead to extraneous solutions.

Q: How do I simplify the absolute value?

A: To simplify the absolute value, you can start by removing the absolute value sign and then squaring both sides of the equation.

Q: Can I use this equation to solve equations with complex numbers?

A: Yes, you can use this equation to solve equations with complex numbers. However, you will need to be careful when working with complex numbers, as they can sometimes lead to extraneous solutions.

Q: How do I simplify the complex number?

A: To simplify the complex number, you can start by removing the imaginary part and then squaring both sides of the equation.

Conclusion

In this article, we addressed some of the most frequently asked questions about the equation $\sqrt{10n + 1} = \sqrt{2n + 1}$. We hope that this article has been helpful in clarifying any confusion you may have had about the equation and its solution. If you have any further questions, please don't hesitate to ask.