Solve The Equation:$\sqrt{8x - 48} - 15 = -11$

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Introduction

Solving equations involving square roots can be a challenging task, especially when they involve variables. In this article, we will focus on solving the equation $\sqrt{8x - 48} - 15 = -11$. This equation requires careful manipulation and application of algebraic techniques to isolate the variable $x$. Our goal is to provide a step-by-step solution to this equation, making it easier for readers to understand and apply the concepts.

Understanding the Equation

The given equation is $\sqrt{8x - 48} - 15 = -11$. To begin solving this equation, we need to isolate the square root term. We can do this by adding 15 to both sides of the equation, which gives us:

$\sqrt{8x - 48} = -11 + 15$

$\sqrt{8x - 48} = 4$

Simplifying the Equation

Now that we have isolated the square root term, we can simplify the equation by squaring both sides. This will help us eliminate the square root and make it easier to solve for $x$. Squaring both sides of the equation gives us:

$8x - 48 = 4^2$

$8x - 48 = 16$

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. To do this, we need to isolate the variable $x$ on one side of the equation. We can do this by adding 48 to both sides of the equation, which gives us:

$8x = 16 + 48$

$8x = 64$

Final Step

Now that we have isolated the variable $x$, we can solve for its value. To do this, we need to divide both sides of the equation by 8, which gives us:

$x = \frac{64}{8}$

$x = 8$

Conclusion

In this article, we have solved the equation $\sqrt{8x - 48} - 15 = -11$. We started by isolating the square root term, then simplified the equation by squaring both sides. Finally, we solved for the value of $x$ by isolating the variable on one side of the equation. The final solution is $x = 8$. This equation requires careful manipulation and application of algebraic techniques to isolate the variable $x$. Our goal is to provide a step-by-step solution to this equation, making it easier for readers to understand and apply the concepts.

Tips and Tricks

• When solving equations involving square roots, it's essential to isolate the square root term first.
• Squaring both sides of the equation can help eliminate the square root and make it easier to solve for the variable.
• Always check your work by plugging the solution back into the original equation to ensure that it's true.

Real-World Applications

Solving equations involving square roots has many real-world applications. For example, in physics, the equation for the time it takes for an object to fall from a certain height involves a square root term. In engineering, the equation for the stress on a beam involves a square root term. By understanding how to solve equations involving square roots, we can apply these concepts to real-world problems and make informed decisions.

Common Mistakes

• Failing to isolate the square root term first can lead to incorrect solutions.
• Squaring both sides of the equation without checking the solution can lead to extraneous solutions.
• Not checking the solution by plugging it back into the original equation can lead to incorrect solutions.

Final Thoughts

Solving equations involving square roots requires careful manipulation and application of algebraic techniques. By following the steps outlined in this article, we can solve equations like $\sqrt{8x - 48} - 15 = -11$ and apply these concepts to real-world problems. Remember to always check your work by plugging the solution back into the original equation to ensure that it's true.

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Introduction

In our previous article, we solved the equation $\sqrt{8x - 48} - 15 = -11$ using algebraic techniques. In this article, we will provide a Q&A section to help readers understand the concepts and techniques used to solve this equation. We will answer common questions and provide additional tips and tricks to help readers master the art of solving equations involving square roots.

Q&A

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 can be done by adding or subtracting the same value to both sides of the equation.

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

A: Squaring both sides of the equation helps to eliminate the square root term and make it easier to solve for the variable. However, it's essential to check the solution by plugging it back into the original equation to ensure that it's true.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid for the original equation. This can occur when we square both sides of the equation without checking the solution.

Q: How can we avoid extraneous solutions?

A: To avoid extraneous solutions, we need to check the solution by plugging it back into the original equation. This ensures that the solution is valid and true for the original equation.

Q: What is the final solution to the equation $\sqrt{8x - 48} - 15 = -11$?

A: The final solution to the equation $\sqrt{8x - 48} - 15 = -11$ is $x = 8$.

Q: Can we use the same techniques to solve other equations involving square roots?

A: Yes, the techniques used to solve the equation $\sqrt{8x - 48} - 15 = -11$ can be applied to other equations involving square roots. However, it's essential to understand the concepts and techniques used to solve these equations.

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:

• Failing to isolate the square root term first
• Squaring both sides of the equation without checking the solution
• Not checking the solution by plugging it back into the original equation

Tips and Tricks

• Always check your work by plugging the solution back into the original equation to ensure that it's true.
• Use algebraic techniques to isolate the square root term and make it easier to solve for the variable.
• Be careful when squaring both sides of the equation, as this can lead to extraneous solutions.

Real-World Applications

Solving equations involving square roots has many real-world applications. For example, in physics, the equation for the time it takes for an object to fall from a certain height involves a square root term. In engineering, the equation for the stress on a beam involves a square root term. By understanding how to solve equations involving square roots, we can apply these concepts to real-world problems and make informed decisions.

Common Mistakes

• Failing to isolate the square root term first can lead to incorrect solutions.
• Squaring both sides of the equation without checking the solution can lead to extraneous solutions.
• Not checking the solution by plugging it back into the original equation can lead to incorrect solutions.

Final Thoughts

Solving equations involving square roots requires careful manipulation and application of algebraic techniques. By following the steps outlined in this article and understanding the concepts and techniques used, we can solve equations like $\sqrt{8x - 48} - 15 = -11$ and apply these concepts to real-world problems. Remember to always check your work by plugging the solution back into the original equation to ensure that it's true.