Solve The Equation 2 X − 1 = 3 \sqrt{2x} - 1 = 3 2 X ​ − 1 = 3 .A) X = 2 X = 2 X = 2 B) X = 16 X = 16 X = 16 C) X = 8 X = 8 X = 8 D) X = 4 X = 4 X = 4

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Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will focus on solving the equation $\sqrt{2x} - 1 = 3$ using algebraic manipulation and square root properties. We will also provide a step-by-step solution to help you understand the process.

Understanding the Equation

The given equation is $\sqrt{2x} - 1 = 3$. Our goal is to isolate the variable $x$ and find its value. To do this, we need to get rid of the square root term. We can do this by adding $1$ to both sides of the equation, which will give us $\sqrt{2x} = 4$.

Isolating the Square Root Term

Now that we have $\sqrt{2x} = 4$, we can square both sides of the equation to get rid of the square root. This will give us $2x = 16$. Squaring both sides of an equation is a common technique used to eliminate square root terms.

Solving for $x$

Now that we have $2x = 16$, we can solve for $x$ by dividing both sides of the equation by $2$. This will give us $x = 8$. Therefore, the value of $x$ that satisfies the equation $\sqrt{2x} - 1 = 3$ is $x = 8$.

Checking the Solution

To verify our solution, we can plug $x = 8$ back into the original equation and check if it is true. Substituting $x = 8$ into the equation $\sqrt{2x} - 1 = 3$, we get $\sqrt{2(8)} - 1 = 3$, which simplifies to $\sqrt{16} - 1 = 3$. Since $\sqrt{16} = 4$, we have $4 - 1 = 3$, which is indeed true. Therefore, our solution $x = 8$ is correct.

Conclusion

In this article, we solved the equation $\sqrt{2x} - 1 = 3$ using algebraic manipulation and square root properties. We added $1$ to both sides of the equation to get rid of the square root term, squared both sides to eliminate the square root, and solved for $x$ by dividing both sides of the equation by $2$. We also verified our solution by plugging $x = 8$ back into the original equation. The value of $x$ that satisfies the equation $\sqrt{2x} - 1 = 3$ is indeed $x = 8$.

Final Answer

The final answer is $\boxed{8}$.

Step-by-Step Solution

Here is a step-by-step solution to the equation $\sqrt{2x} - 1 = 3$:

1. Add $1$ to both sides of the equation to get rid of the square root term: $\sqrt{2x} = 4$
2. Square both sides of the equation to eliminate the square root: $2x = 16$
3. Divide both sides of the equation by $2$ to solve for $x$: $x = 8$

Common Mistakes to Avoid

When solving equations involving square roots, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not adding $1$ to both sides of the equation to get rid of the square root term
• Not squaring both sides of the equation to eliminate the square root
• Not dividing both sides of the equation by $2$ to solve for $x$

Tips and Tricks

Here are some tips and tricks to help you solve equations involving square roots:

• Always add $1$ to both sides of the equation to get rid of the square root term
• Always square both sides of the equation to eliminate the square root
• Always divide both sides of the equation by $2$ to solve for $x$

Real-World Applications

Solving equations involving square roots has many real-world applications. Here are a few examples:

• Physics: When solving problems involving motion, you may need to use equations involving square roots to find the velocity or acceleration of an object.
• Engineering: When designing structures, you may need to use equations involving square roots to find the stress or strain on a material.
• Computer Science: When writing algorithms, you may need to use equations involving square roots to find the distance between two points in a graph.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we solved the equation $\sqrt{2x} - 1 = 3$ using algebraic manipulation and square root properties. We added $1$ to both sides of the equation to get rid of the square root term, squared both sides to eliminate the square root, and solved for $x$ by dividing both sides of the equation by $2$. We also verified our solution by plugging $x = 8$ back into the original equation. The value of $x$ that satisfies the equation $\sqrt{2x} - 1 = 3$ is indeed $x = 8$.

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will provide a Q&A section to help you understand the process of solving equations involving square roots. We will cover common questions and provide step-by-step solutions to help you grasp the concept.

Q1: What is the first step in solving an equation involving a square root?

A1: The first step in solving an equation involving a square root is to isolate the square root term. This can be done by adding or subtracting a constant to both sides of the equation.

Q2: How do I get rid of the square root term in an equation?

A2: To get rid of the square root term in an equation, you can square both sides of the equation. This will eliminate the square root and allow you to solve for the variable.

Q3: What is the difference between squaring and taking the square root?

A3: Squaring a number means multiplying it by itself, while taking the square root of a number means finding the value that, when multiplied by itself, gives the original number.

Q4: How do I solve for x in an equation involving a square root?

A4: To solve for x in an equation involving a square root, you can follow these steps:

1. Isolate the square root term by adding or subtracting a constant to both sides of the equation.
2. Square both sides of the equation to eliminate the square root.
3. Solve for x by dividing both sides of the equation by the coefficient of x.

Q5: What is the final step in solving an equation involving a square root?

A5: The final step in solving an equation involving a square root is to check your solution by plugging it back into the original equation. This will ensure that your solution is correct.

Q6: Can I use a calculator to solve equations involving square roots?

A6: Yes, you can use a calculator to solve equations involving square roots. However, it's always a good idea to check your solution by plugging it back into the original equation to ensure that it's correct.

Q7: What are some common mistakes to avoid when solving equations involving square roots?

A7: Some common mistakes to avoid when solving equations involving square roots include:

• Not isolating the square root term
• Not squaring both sides of the equation
• Not checking your solution by plugging it back into the original equation

Q8: How do I apply the concept of solving equations involving square roots to real-world problems?

A8: The concept of solving equations involving square roots can be applied to a variety of real-world problems, including:

• Physics: When solving problems involving motion, you may need to use equations involving square roots to find the velocity or acceleration of an object.
• Engineering: When designing structures, you may need to use equations involving square roots to find the stress or strain on a material.
• Computer Science: When writing algorithms, you may need to use equations involving square roots to find the distance between two points in a graph.

Q9: What are some tips and tricks for solving equations involving square roots?

A9: Some tips and tricks for solving equations involving square roots include:

• Always isolate the square root term
• Always square both sides of the equation
• Always check your solution by plugging it back into the original equation

Q10: How do I know if my solution is correct?

A10: To know if your solution is correct, you can plug it back into the original equation and check if it's true. If it's true, then your solution is correct.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right approach, it can be done efficiently. In this Q&A section, we provided step-by-step solutions to common questions and covered tips and tricks for solving equations involving square roots. We also discussed real-world applications of the concept and provided a final check to ensure that your solution is correct.