What Is The Solution Of X − 4 + 5 = 2 \sqrt{x-4} + 5 = 2 X − 4 ​ + 5 = 2 ?A. X = − 17 X = -17 X = − 17 B. X = 13 X = 13 X = 13 C. X = 53 X = 53 X = 53 D. No Solution

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Introduction

Solving equations involving square roots can be a challenging task, especially when the equation is not in a straightforward form. In this article, we will explore the solution to the equation $\sqrt{x-4} + 5 = 2$ and determine whether it has a solution or not.

Understanding the Equation

The given equation is $\sqrt{x-4} + 5 = 2$. To solve this equation, we need to isolate the square root term. The first step is to subtract 5 from both sides of the equation, which gives us $\sqrt{x-4} = -3$.

Isolating the Square Root Term

Now that we have isolated the square root term, we need to get rid of the square root. To do this, we can square both sides of the equation. However, we need to be careful when squaring both sides, as it can introduce extraneous solutions.

Squaring Both Sides

Squaring both sides of the equation $\sqrt{x-4} = -3$ gives us $x-4 = (-3)^2$. Simplifying the right-hand side, we get $x-4 = 9$.

Solving for x

Now that we have $x-4 = 9$, we can solve for x by adding 4 to both sides of the equation. This gives us $x = 13$.

Checking the Solution

Before we conclude that $x = 13$ is the solution to the equation, we need to check if it satisfies the original equation. Substituting $x = 13$ into the original equation, we get $\sqrt{13-4} + 5 = \sqrt{9} + 5 = 3 + 5 = 8$. However, the original equation states that $\sqrt{x-4} + 5 = 2$, which means that $\sqrt{x-4} = -3$. Since $\sqrt{9} = 3$, not $-3$, we have found an extraneous solution.

Conclusion

Since $x = 13$ is an extraneous solution, we need to re-examine the equation and determine if it has a solution or not. Going back to the equation $\sqrt{x-4} = -3$, we can see that the square root of a number cannot be negative. This means that the equation has no solution.

Final Answer

The final answer is D. No solution.

Discussion

The equation $\sqrt{x-4} + 5 = 2$ may seem like a simple equation, but it can be deceiving. The presence of the square root term makes it challenging to solve, and the introduction of extraneous solutions can be frustrating. However, by carefully following the steps and checking the solution, we can determine that the equation has no solution.

Common Mistakes

When solving equations involving square roots, it's essential to be careful when squaring both sides. This can introduce extraneous solutions, which can be misleading. Additionally, it's crucial to check the solution to ensure that it satisfies the original equation.

Tips and Tricks

When solving equations involving square roots, it's helpful to:

• Isolate the square root term
• Square both sides carefully
• Check the solution to ensure that it satisfies the original equation
• Be aware of extraneous solutions

By following these tips and tricks, you can increase your chances of solving equations involving square roots correctly.

Related Topics

If you're interested in learning more about solving equations involving square roots, you may want to explore the following topics:

• Solving quadratic equations
• Solving equations involving absolute values
• Solving equations involving fractions

These topics can help you build a stronger foundation in algebra and prepare you for more advanced math concepts.

Conclusion

Solving equations involving square roots can be a challenging task, but with careful attention to detail and a thorough understanding of the steps involved, you can determine whether the equation has a solution or not. In this article, we explored the solution to the equation $\sqrt{x-4} + 5 = 2$ and found that it has no solution. By following the tips and tricks outlined in this article, you can increase your chances of solving equations involving square roots correctly.

Introduction

Solving equations involving square roots can be a challenging task, but with the right approach and techniques, you can master it. In this article, we will answer some of the most frequently asked questions about 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. This means getting the square root term by itself on one side of the equation.

Q: How do I isolate the square root term?

A: To isolate the square root term, you can add or subtract the same value to both sides of the equation. For example, if the equation is $\sqrt{x-4} + 5 = 2$, you can subtract 5 from both sides to get $\sqrt{x-4} = -3$.

Q: What happens when I square both sides of the equation?

A: When you square both sides of the equation, you are essentially getting rid of the square root. However, you need to be careful because squaring both sides can introduce extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to substitute the solution back into the original equation and see if it satisfies the equation. If it doesn't, then it's an extraneous solution.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that satisfies the equation after squaring both sides, but it doesn't satisfy the original equation.

Q: How do I avoid extraneous solutions?

A: To avoid extraneous solutions, you need to be careful when squaring both sides of the equation. Make sure to check the solution by substituting it back into the original equation.

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

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

• Squaring both sides of the equation without checking the solution
• Not isolating the square root term
• Not checking for extraneous solutions

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

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

• Isolating the square root term
• Squaring both sides carefully
• Checking the solution by substituting it back into the original equation
• Being aware of extraneous solutions

Q: Can you give an example of an equation involving a square root that has no solution?

A: Yes, an example of an equation involving a square root that has no solution is $\sqrt{x-4} = -3$. Since the square root of a number cannot be negative, this equation has no solution.

Q: Can you give an example of an equation involving a square root that has multiple solutions?

A: Yes, an example of an equation involving a square root that has multiple solutions is $\sqrt{x-4} = 3$. This equation has two solutions: $x = 13$ and $x = 7$.

Q: How do I determine if an equation involving a square root has a solution or not?

A: To determine if an equation involving a square root has a solution or not, you need to check if the square root term is positive or negative. If it's positive, then the equation has a solution. If it's negative, then the equation has no solution.

Conclusion

Solving equations involving square roots can be a challenging task, but with the right approach and techniques, you can master it. By following the tips and tricks outlined in this article, you can increase your chances of solving equations involving square roots correctly. Remember to isolate the square root term, square both sides carefully, and check the solution by substituting it back into the original equation.