Given The Equation 3 X + 4 − 10 = 8 3 \sqrt{x+4} - 10 = 8 3 X + 4 ​ − 10 = 8 , State The Solution(s). If A Possible Solution Was Extraneous, Type extraneous In The Second Answer Box. X = □ X = \square X = □ □ \square □

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Introduction

Solving equations involving square roots can be challenging, but with the right approach, we can find the solution(s) to the given equation $3 \sqrt{x+4} - 10 = 8$. In this article, we will walk through the steps to isolate the variable $x$ and find the possible solution(s).

Step 1: Isolate the Square Root Term

The first step is to isolate the square root term on one side of the equation. We can do this by adding 10 to both sides of the equation:

$$3 \sqrt{x+4} = 8 + 10$$

This simplifies to:

$$3 \sqrt{x+4} = 18$$

Step 2: Divide by 3

Next, we divide both sides of the equation by 3 to isolate the square root term:

$$\sqrt{x+4} = \frac{18}{3}$$

This simplifies to:

$$\sqrt{x+4} = 6$$

Step 3: Square Both Sides

To eliminate the square root, we square both sides of the equation:

$$(\sqrt{x+4})^2 = 6^2$$

This simplifies to:

$$x + 4 = 36$$

Step 4: Solve for x

Now, we can solve for $x$ by subtracting 4 from both sides of the equation:

$$x = 36 - 4$$

This simplifies to:

$$x = 32$$

Step 2: Check for Extraneous Solutions

Before we can confirm that $x = 32$ is a valid solution, we need to check if it satisfies the original equation. We can do this by plugging $x = 32$ back into the original equation:

$$3 \sqrt{32+4} - 10 = 3 \sqrt{36} - 10$$

This simplifies to:

$$3 \sqrt{36} - 10 = 3(6) - 10$$

This simplifies to:

$$18 - 10 = 8$$

This is true, so $x = 32$ is a valid solution.

Conclusion

In this article, we solved the equation $3 \sqrt{x+4} - 10 = 8$ and found the solution $x = 32$. We also checked for extraneous solutions and confirmed that $x = 32$ is a valid solution.

Final Answer

The final answer is: $\boxed{32}$

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Introduction

In our previous article, we solved the equation $3 \sqrt{x+4} - 10 = 8$ and found the solution $x = 32$. 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 term on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation.

Q: How do I know if a solution is extraneous?

A: A solution is extraneous if it does not satisfy the original equation. To check if a solution is extraneous, plug the solution back into the original equation and simplify. If the equation is not true, then the solution is extraneous.

Q: What is the difference between a valid solution and an extraneous solution?

A: A valid solution is a solution that satisfies the original equation, while an extraneous solution is a solution that does not satisfy the original equation.

Q: Can I use the same steps to solve an equation involving a cube root?

A: No, the steps to solve an equation involving a cube root are different from the steps to solve an equation involving a square root. To solve an equation involving a cube root, you need to isolate the cube root term and then cube both sides of the equation.

Q: How do I know if an equation involving a square root has a solution?

A: If the equation involves a square root, it may have a solution if the expression inside the square root is non-negative. If the expression inside the square root is negative, then the equation has no solution.

Q: Can I use a calculator to solve an equation involving a square root?

A: Yes, you can use a calculator to solve an equation involving a square root. However, you need to be careful when using a calculator to check for extraneous solutions.

Q: What is the most common mistake when solving equations involving square roots?

A: The most common mistake when solving equations involving square roots is to forget to check for extraneous solutions.

Q: Can I use the same steps to solve an equation involving a square root and a linear term?

A: Yes, you can use the same steps to solve an equation involving a square root and a linear term. However, you need to be careful when isolating the square root term.

Q: How do I know if an equation involving a square root and a linear term has a solution?

A: If the equation involves a square root and a linear term, it may have a solution if the expression inside the square root is non-negative and the linear term is equal to zero.

Conclusion

In this article, we answered some common questions related to solving equations involving square roots. We hope that this article has been helpful in clarifying some of the common misconceptions and mistakes when solving equations involving square roots.

Final Answer

The final answer is: $\boxed{32}$