Question 3Solve 30 − N = N \sqrt{30-n}=n 30 − N ​ = N A. N = 5 N=5 N = 5 And N = − 6 N=-6 N = − 6 B. N = 5 N=5 N = 5 C. No SolutionD. N = − 6 N=-6 N = − 6

$$

Introduction

In this article, we will be solving the equation $\sqrt{30-n}=n$. This equation involves a square root and a variable $n$, and we need to find the value of $n$ that satisfies the equation. We will use algebraic techniques to solve the equation and find the possible values of $n$.

Step 1: Square Both Sides

The first step in solving the equation is to square both sides of the equation. This will eliminate the square root and give us a quadratic equation in terms of $n$. We have:

$$\sqrt{30-n}=n$$

Squaring both sides of the equation, we get:

$$30-n=n^2$$

Step 2: Rearrange the Equation

Next, we need to rearrange the equation to get all the terms on one side. We can do this by subtracting $n^2$ from both sides of the equation:

$$30-n-n^2=0$$

Step 3: Factor the Quadratic

The equation is now a quadratic equation in terms of $n$. We can factor the quadratic to get:

$$(n+6)(n-5)=0$$

Step 4: Solve for $n$

Now that we have factored the quadratic, we can solve for $n$ by setting each factor equal to zero:

$$n+6=0 \quad \text{or} \quad n-5=0$$

Solving for $n$, we get:

$$n=-6 \quad \text{or} \quad n=5$$

Conclusion

We have solved the equation $\sqrt{30-n}=n$ and found two possible values of $n$: $n=-6$ and $n=5$. We can check these values by plugging them back into the original equation to see if they satisfy the equation.

Checking the Solutions

Let's check the solutions by plugging them back into the original equation:

$$\sqrt{30-(-6)}=-6 \quad \text{or} \quad \sqrt{30-5}=5$$

Simplifying the expressions, we get:

$$\sqrt{36}=-6 \quad \text{or} \quad \sqrt{25}=5$$

Since $\sqrt{36}=6$ and $\sqrt{25}=5$, we can see that only $n=5$ satisfies the original equation.

Final Answer

The final answer is $\boxed{5}$.

Discussion

The equation $\sqrt{30-n}=n$ is a quadratic equation in terms of $n$. We solved the equation by squaring both sides, rearranging the equation, factoring the quadratic, and solving for $n$. We found two possible values of $n$: $n=-6$ and $n=5$. However, only $n=5$ satisfies the original equation.

Related Topics

• Solving quadratic equations
• Factoring quadratics
• Square roots and algebraic expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

$$ $$

Introduction

In our previous article, we solved the equation $\sqrt{30-n}=n$ and found two possible values of $n$: $n=-6$ and $n=5$. However, only $n=5$ satisfies the original equation. In this article, we will answer some common questions related to the equation and provide additional information to help you understand the solution.

Q: What is the equation $\sqrt{30-n}=n$?

A: The equation $\sqrt{30-n}=n$ is a quadratic equation in terms of $n$. It involves a square root and a variable $n$, and we need to find the value of $n$ that satisfies the equation.

Q: How do I solve the equation $\sqrt{30-n}=n$?

A: To solve the equation, we need to square both sides of the equation, rearrange the equation, factor the quadratic, and solve for $n$. We can use the following steps:

1. Square both sides of the equation: $\sqrt{30-n}=n$
2. Rearrange the equation: $30-n=n^2$
3. Factor the quadratic: $(n+6)(n-5)=0$
4. Solve for $n$: $n=-6$ or $n=5$

Q: Why do I get two possible values of $n$: $n=-6$ and $n=5$?

A: We get two possible values of $n$ because the equation is a quadratic equation, and it has two solutions. However, only one of the solutions satisfies the original equation.

Q: How do I check if the solutions are correct?

A: To check if the solutions are correct, we need to plug them back into the original equation and see if they satisfy the equation. We can use the following steps:

1. Plug $n=-6$ back into the original equation: $\sqrt{30-(-6)}=-6$
2. Simplify the expression: $\sqrt{36}=-6$
3. Check if the expression is true: $\sqrt{36}=6$, not $-6$

Since the expression is not true, we can conclude that $n=-6$ is not a solution to the equation.

Q: What if I get a negative value for $n$?

A: If you get a negative value for $n$, it means that the solution is extraneous. This is because the square root of a negative number is not a real number.

Q: Can I use other methods to solve the equation?

A: Yes, you can use other methods to solve the equation, such as using the quadratic formula or graphing the equation. However, the method we used in this article is the most straightforward and easiest to understand.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

• Squaring both sides of the equation without checking if the expression is true
• Not checking if the solutions are correct
• Not considering the possibility of extraneous solutions

Conclusion

In this article, we answered some common questions related to the equation $\sqrt{30-n}=n$ and provided additional information to help you understand the solution. We also discussed some common mistakes to avoid when solving the equation. By following the steps and tips provided in this article, you should be able to solve the equation and understand the solution.

Related Topics

• Solving quadratic equations
• Factoring quadratics
• Square roots and algebraic expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Note: The references provided are for general information and are not specific to the equation $\sqrt{30-n}=n$.