Solve The Equation For { X $} : : : { \sqrt{x+7} = 9 \}

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Introduction

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Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will focus on solving the equation $\sqrt{x+7} = 9$ for the variable $x$. We will use a step-by-step approach to ensure that we understand the solution and can apply it to similar problems.

Understanding the Equation

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The given equation is $\sqrt{x+7} = 9$. This equation involves a square root, which means that the expression inside the square root must be non-negative. In this case, the expression $x+7$ must be greater than or equal to zero.

Why is this important?

If $x+7$ is negative, then the square root of $x+7$ would be an imaginary number, which is not a real solution to the equation. Therefore, we must ensure that $x+7 \geq 0$ before proceeding with the solution.

Step 1: Isolate the Square Root

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The first step in solving the equation is to isolate the square root. In this case, we can do this by squaring both sides of the equation.

Squaring both sides

$\left(\sqrt{x+7}\right)^2 = 9^2$

This simplifies to:

$x+7 = 81$

Step 2: Solve for x

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Now that we have isolated the square root, we can solve for $x$ by subtracting 7 from both sides of the equation.

Subtracting 7 from both sides

$x = 81 - 7$

This simplifies to:

$x = 74$

Conclusion

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In this article, we solved the equation $\sqrt{x+7} = 9$ for the variable $x$. We used a step-by-step approach to ensure that we understood the solution and could apply it to similar problems. By isolating the square root and then solving for $x$, we found that $x = 74$.

What's next?

If you have any questions or need further clarification on this topic, feel free to ask. We will be happy to help. Additionally, if you would like to learn more about solving equations involving square roots, we have a comprehensive guide available.

Frequently Asked Questions

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Q: What if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the square root would be an imaginary number, which is not a real solution to the equation.

Q: Can I use this method to solve other equations involving square roots?

A: Yes, this method can be applied to solve other equations involving square roots. However, you must ensure that the expression inside the square root is non-negative.

Q: What if I get a negative value for x?

A: If you get a negative value for $x$, then it means that the expression inside the square root is negative, and the square root would be an imaginary number.

Additional Resources

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If you would like to learn more about solving equations involving square roots, we recommend the following resources:

• Solving Equations Involving Square Roots
• Square Root Properties

Final Thoughts

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Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. By isolating the square root and then solving for $x$, we can find the solution to the equation. We hope that this article has provided you with a clear understanding of how to solve equations involving square roots.

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Introduction

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In our previous article, we solved the equation $\sqrt{x+7} = 9$ for the variable $x$. We used a step-by-step approach to ensure that we understood the solution and could apply it to similar problems. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.

Q&A Guide

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Q: What is the first step in solving the equation $\sqrt{x+7} = 9$?

A: The first step in solving the equation is to isolate the square root. In this case, we can do this by squaring both sides of the equation.

Q: Why do we need to square both sides of the equation?

A: Squaring both sides of the equation allows us to eliminate the square root and solve for $x$. This is because the square of a square root is equal to the expression inside the square root.

Q: What happens if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the square root would be an imaginary number, which is not a real solution to the equation.

Q: Can I use this method to solve other equations involving square roots?

A: Yes, this method can be applied to solve other equations involving square roots. However, you must ensure that the expression inside the square root is non-negative.

Q: What if I get a negative value for $x$?

A: If you get a negative value for $x$, then it means that the expression inside the square root is negative, and the square root would be an imaginary number.

Q: How do I know if the expression inside the square root is non-negative?

A: To determine if the expression inside the square root is non-negative, you can check if the expression is greater than or equal to zero. In this case, we have $x+7 \geq 0$, which means that $x \geq -7$.

Q: Can I use this method to solve equations involving square roots with different bases?

A: Yes, this method can be applied to solve equations involving square roots with different bases. However, you must ensure that the expression inside the square root is non-negative.

Q: What if I have an equation involving a square root with a coefficient?

A: If you have an equation involving a square root with a coefficient, you can multiply both sides of the equation by the coefficient to eliminate it. For example, if you have the equation $\sqrt{3x+7} = 9$, you can multiply both sides by 3 to get $3\sqrt{x+7} = 27$.

Common Mistakes

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Mistake 1: Not checking if the expression inside the square root is non-negative

A: Failing to check if the expression inside the square root is non-negative can lead to incorrect solutions.

Mistake 2: Not squaring both sides of the equation

A: Failing to square both sides of the equation can lead to incorrect solutions.

Mistake 3: Not checking if the solution is valid

A: Failing to check if the solution is valid can lead to incorrect solutions.

Conclusion

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In this article, we provided a Q&A guide to help you better understand the solution to the equation $\sqrt{x+7} = 9$. We addressed common questions and mistakes that can occur when solving equations involving square roots. By following the steps outlined in this article, you can ensure that you are solving equations involving square roots correctly.

Additional Resources

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If you would like to learn more about solving equations involving square roots, we recommend the following resources:

• Solving Equations Involving Square Roots
• Square Root Properties

Final Thoughts

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Solving equations involving square roots can be a challenging task, but with the right approach, it can be broken down into manageable steps. By following the steps outlined in this article, you can ensure that you are solving equations involving square roots correctly.