What Is The Solution To The Equation Below?$\[4+\sqrt{6x+4}=8\\]A. \[$x=-\frac{10}{3}\$\]B. \[$x=-\frac{1}{3}\$\]C. \[$x=2\$\]D. \[$x=16\$\]

Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. In this article, we will focus on solving a specific equation, which is given below:

${4+\sqrt{6x+4}=8}$

Our goal is to find the value of x that satisfies this equation. We will use algebraic techniques to isolate the variable x and find its value.

Step 1: Isolate the Square Root

The first step in solving this equation is to isolate the square root term. We can do this by subtracting 4 from both sides of the equation:

${4+\sqrt{6x+4}-4=8-4}$

This simplifies to:

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

Step 2: Square Both Sides

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

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

This simplifies to:

${6x+4=16}$

Step 3: Isolate the Variable

Now, we need to isolate the variable x. We can do this by subtracting 4 from both sides of the equation:

${6x+4-4=16-4}$

This simplifies to:

${6x=12}$

Step 4: Solve for x

Finally, we can solve for x by dividing both sides of the equation by 6:

${6x/6=12/6}$

This simplifies to:

${x=2}$

Conclusion

In this article, we solved the equation:

${4+\sqrt{6x+4}=8}$

Using algebraic techniques, we isolated the variable x and found its value to be x = 2. This is the correct solution to the equation.

Comparison with Answer Choices

Let's compare our solution with the answer choices:

A. ${x=-\frac{10}{3}}$ B. ${x=-\frac{1}{3}}$ C. ${x=2}$ D. ${x=16}$

Our solution, x = 2, matches answer choice C. Therefore, the correct answer is:

$$

Q: What is the first step in solving the equation 4+\sqrt{6x+4}=8?

A: The first step in solving this equation is to isolate the square root term. We can do this by subtracting 4 from both sides of the equation:

${4+\sqrt{6x+4}-4=8-4}$

This simplifies to:

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

Q: Why do we need to isolate the square root term?

A: We need to isolate the square root term because it is the only term that contains the variable x. By isolating the square root term, we can eliminate it and solve for x.

Q: What is the next step in solving the equation?

A: The next step in solving this equation is to square both sides of the equation:

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

This simplifies to:

${6x+4=16}$

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 root term. This allows us to solve for x.

Q: How do we isolate the variable x?

A: We can isolate the variable x by subtracting 4 from both sides of the equation:

${6x+4-4=16-4}$

This simplifies to:

${6x=12}$

Q: What is the final step in solving the equation?

A: The final step in solving this equation is to solve for x by dividing both sides of the equation by 6:

${6x/6=12/6}$

This simplifies to:

${x=2}$

Q: Why is it important to check our work?

A: It is essential to check our work to ensure that we have found the correct solution to the equation. In this case, we can check our work by plugging x = 2 back into the original equation:

${4+\sqrt{6(2)+4}=8}$

This simplifies to:

${4+\sqrt{16}=8}$

Which is true. Therefore, we can be confident that our solution is correct.

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

A: Some common mistakes to avoid when solving equations include:

• Not isolating the variable x
• Not squaring both sides of the equation
• Not checking our work
• Not following the order of operations

By avoiding these common mistakes, we can ensure that we find the correct solution to the equation.

Q: How can we apply the skills we learned in this article to real-world problems?

A: The skills we learned in this article can be applied to a wide range of real-world problems, including:

• Solving equations in physics and engineering
• Modeling population growth and decay
• Analyzing financial data
• Solving optimization problems

By applying the skills we learned in this article, we can solve complex problems and make informed decisions in a variety of fields.