Solve For \[$ K \$\]:$\[ \sqrt{-9k} = \sqrt{16-k} \\]\[$ K = \$\]

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Introduction

Solving equations involving square roots can be a challenging task, especially when dealing with variables inside the square root. In this problem, we are given an equation with a square root on both sides, and we need to solve for the variable $k$. The equation is $\sqrt{-9k} = \sqrt{16-k}$, and our goal is to isolate the variable $k$ and find its value.

Step 1: Square Both Sides of the Equation

To eliminate the square roots, we can square both sides of the equation. This will allow us to simplify the equation and make it easier to solve for $k$. Squaring both sides of the equation gives us:

$$(-9k) = (16-k)$$

Step 2: Expand and Simplify the Equation

Now that we have squared both sides of the equation, we can expand and simplify it. Expanding the right-hand side of the equation gives us:

$$-9k = 16 - k$$

Step 3: Add $k$ to Both Sides of the Equation

To get all the terms with $k$ on one side of the equation, we can add $k$ to both sides of the equation. This gives us:

$$-9k + k = 16 - k + k$$

Step 4: Simplify the Equation

Now that we have added $k$ to both sides of the equation, we can simplify it. The left-hand side of the equation simplifies to:

$$-8k = 16$$

Step 5: Divide Both Sides of the Equation by $-8$

To solve for $k$, we need to isolate the variable. We can do this by dividing both sides of the equation by $-8$. This gives us:

$$k = \frac{16}{-8}$$

Step 6: Simplify the Fraction

Now that we have divided both sides of the equation by $-8$, we can simplify the fraction. The fraction $\frac{16}{-8}$ simplifies to:

$$k = -2$$

Conclusion

In this problem, we were given an equation with a square root on both sides, and we needed to solve for the variable $k$. We started by squaring both sides of the equation to eliminate the square roots, and then we expanded and simplified the equation. We added $k$ to both sides of the equation to get all the terms with $k$ on one side, and then we simplified the equation. Finally, we divided both sides of the equation by $-8$ to solve for $k$. The value of $k$ is $-2$.

Final Answer

The final answer is $\boxed{-2}$.

Discussion

Solving equations involving square roots can be a challenging task, but with the right steps, we can simplify the equation and make it easier to solve. In this problem, we used the steps of squaring both sides of the equation, expanding and simplifying the equation, adding $k$ to both sides of the equation, and dividing both sides of the equation by $-8$ to solve for $k$. These steps can be applied to other problems involving square roots, and with practice, we can become more confident in our ability to solve these types of equations.

Related Problems

If you are looking for more practice problems involving square roots, here are a few related problems:

• $\sqrt{25-x} = \sqrt{x-9}$
• $\sqrt{9x} = \sqrt{x+16}$
• $\sqrt{x-4} = \sqrt{16-x}$

These problems involve square roots and require the same steps as the original problem to solve. With practice, you can become more confident in your ability to solve these types of equations.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right steps, we can simplify the equation and make it easier to solve. In this problem, we used the steps of squaring both sides of the equation, expanding and simplifying the equation, adding $k$ to both sides of the equation, and dividing both sides of the equation by $-8$ to solve for $k$. These steps can be applied to other problems involving square roots, and with practice, we can become more confident in our ability to solve these types of equations.

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Introduction

In our previous article, we solved the equation $\sqrt{-9k} = \sqrt{16-k}$ for the variable $k$. We used the steps of squaring both sides of the equation, expanding and simplifying the equation, adding $k$ to both sides of the equation, and dividing both sides of the equation by $-8$ to solve for $k$. The value of $k$ is $-2$. In this article, we will answer some common questions related to this problem.

Q&A

Q: What is the first step in solving the equation $\sqrt{-9k} = \sqrt{16-k}$?

A: The first step in solving the equation $\sqrt{-9k} = \sqrt{16-k}$ is to square both sides of the equation. This will eliminate the square roots and make it easier to solve for $k$.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the square roots. This is because the square root of a number is equal to the number raised to the power of 1/2. By squaring both sides of the equation, we can get rid of the square roots and make it easier to solve for $k$.

Q: What happens if we don't square both sides of the equation?

A: If we don't square both sides of the equation, we will be left with an equation that contains square roots. This can make it difficult to solve for $k$, and we may end up with multiple solutions or no solution at all.

Q: How do we simplify the equation after squaring both sides?

A: After squaring both sides of the equation, we can simplify it by expanding and combining like terms. This will make it easier to solve for $k$.

Q: What is the final step in solving the equation $\sqrt{-9k} = \sqrt{16-k}$?

A: The final step in solving the equation $\sqrt{-9k} = \sqrt{16-k}$ is to divide both sides of the equation by $-8$. This will give us the value of $k$, which is $-2$.

Q: Why do we need to divide both sides of the equation by $-8$?

A: We need to divide both sides of the equation by $-8$ to solve for $k$. This is because we have $-8k$ on the left-hand side of the equation, and we need to isolate $k$ to find its value.

Q: What if the equation has multiple solutions?

A: If the equation has multiple solutions, we will need to use additional steps to find all the possible values of $k$. This may involve using algebraic techniques such as factoring or using the quadratic formula.

Q: Can we use the same steps to solve other equations involving square roots?

A: Yes, we can use the same steps to solve other equations involving square roots. The steps of squaring both sides of the equation, expanding and simplifying the equation, adding $k$ to both sides of the equation, and dividing both sides of the equation by $-8$ can be applied to other problems involving square roots.

Conclusion

In this article, we answered some common questions related to the equation $\sqrt{-9k} = \sqrt{16-k}$. We discussed the steps involved in solving the equation, including squaring both sides of the equation, expanding and simplifying the equation, adding $k$ to both sides of the equation, and dividing both sides of the equation by $-8$. We also discussed what happens if we don't square both sides of the equation and how to simplify the equation after squaring both sides. With practice, you can become more confident in your ability to solve equations involving square roots.

Final Answer

The final answer is $\boxed{-2}$.

Related Problems

If you are looking for more practice problems involving square roots, here are a few related problems:

• $\sqrt{25-x} = \sqrt{x-9}$
• $\sqrt{9x} = \sqrt{x+16}$
• $\sqrt{x-4} = \sqrt{16-x}$

These problems involve square roots and require the same steps as the original problem to solve. With practice, you can become more confident in your ability to solve these types of equations.