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Introduction

Radical equations are a type of algebraic equation that involves a square root or other radical expression. Solving radical equations requires a combination of algebraic and analytical techniques. In this article, we will focus on solving a specific radical equation, $\sqrt{x+9} + 7 = 9$, and provide a step-by-step guide on how to solve it.

Understanding Radical Equations

A radical equation is an equation that contains a square root or other radical expression. The general form of a radical equation is:

$$\sqrt{x} = a$$

where $a$ is a constant. Radical equations can be solved using various techniques, including algebraic manipulation and analytical methods.

Solving the Radical Equation

The given radical equation is:

$$\sqrt{x+9} + 7 = 9$$

To solve this equation, we need to isolate the square root expression. We can do this by subtracting 7 from both sides of the equation:

$$\sqrt{x+9} = 9 - 7$$

$$\sqrt{x+9} = 2$$

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

$$(\sqrt{x+9})^2 = 2^2$$

$$x+9 = 4$$

Now, we can solve for $x$ by subtracting 9 from both sides of the equation:

$$x = 4 - 9$$

$$x = -5$$

Checking the Solution

To verify that the solution is correct, we can substitute $x = -5$ back into the original equation:

$$\sqrt{x+9} + 7 = 9$$

$$\sqrt{-5+9} + 7 = 9$$

$$\sqrt{4} + 7 = 9$$

$$2 + 7 = 9$$

$$9 = 9$$

Since the equation holds true, we can conclude that the solution is correct.

Conclusion

Solving radical equations requires a combination of algebraic and analytical techniques. By following the steps outlined in this article, we can solve the radical equation $\sqrt{x+9} + 7 = 9$ and find the solution $x = -5$. It's essential to check the solution to ensure that it satisfies the original equation.

Common Mistakes to Avoid

When solving radical equations, it's essential to avoid common mistakes such as:

• Not isolating the square root expression
• Not squaring both sides of the equation
• Not checking the solution

By following the steps outlined in this article and avoiding common mistakes, we can ensure that we solve radical equations accurately and efficiently.

Real-World Applications

Radical equations have numerous real-world applications in various fields, including:

• Physics: Radical equations are used to model the motion of objects and describe the behavior of physical systems.
• Engineering: Radical equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Radical equations are used in algorithms and data structures to solve problems efficiently.

Conclusion

In conclusion, solving radical equations requires a combination of algebraic and analytical techniques. By following the steps outlined in this article, we can solve the radical equation $\sqrt{x+9} + 7 = 9$ and find the solution $x = -5$. It's essential to check the solution to ensure that it satisfies the original equation. By understanding and applying the concepts outlined in this article, we can solve radical equations accurately and efficiently.

Final Answer

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Introduction

Radical equations are a type of algebraic equation that involves a square root or other radical expression. Solving radical equations requires a combination of algebraic and analytical techniques. In this article, we will provide a Q&A guide on solving radical equations, including common mistakes to avoid and real-world applications.

Q: What is a radical equation?

A: A radical equation is an equation that contains a square root or other radical expression. The general form of a radical equation is:

$$\sqrt{x} = a$$

where $a$ is a constant.

Q: How do I solve a radical equation?

A: To solve a radical equation, you need to isolate the square root expression and then square both sides of the equation to eliminate the square root. Here's a step-by-step guide:

1. Isolate the square root expression
2. Square both sides of the equation
3. Solve for the variable

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

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

• Not isolating the square root expression
• Not squaring both sides of the equation
• Not checking the solution

Q: How do I check the solution to a radical equation?

A: To check the solution to a radical equation, you need to substitute the solution back into the original equation and verify that it holds true.

Q: What are some real-world applications of radical equations?

A: Radical equations have numerous real-world applications in various fields, including:

• Physics: Radical equations are used to model the motion of objects and describe the behavior of physical systems.
• Engineering: Radical equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Radical equations are used in algorithms and data structures to solve problems efficiently.

Q: Can you provide an example of a radical equation?

A: Here's an example of a radical equation:

$$\sqrt{x+9} + 7 = 9$$

To solve this equation, you need to isolate the square root expression and then square both sides of the equation.

Q: How do I solve the equation $\sqrt{x+9} + 7 = 9$?

A: To solve the equation $\sqrt{x+9} + 7 = 9$, you need to follow these steps:

1. Isolate the square root expression: $\sqrt{x+9} = 9 - 7$
2. Square both sides of the equation: $(\sqrt{x+9})^2 = 2^2$
3. Solve for the variable: $x+9 = 4$

Q: What is the solution to the equation $\sqrt{x+9} + 7 = 9$?

A: The solution to the equation $\sqrt{x+9} + 7 = 9$ is $x = -5$.

Conclusion

Solving radical equations requires a combination of algebraic and analytical techniques. By following the steps outlined in this article and avoiding common mistakes, we can ensure that we solve radical equations accurately and efficiently. Radical equations have numerous real-world applications in various fields, including physics, engineering, and computer science.

Final Answer

The final answer is $\boxed{-5}$.