Determine Whether The Given Solutions Are Extraneous Or True Solutions For The Following Equations:1. The Solution $w = 8$ Is An Extraneous Solution To The Equation $\sqrt{2w} = -4$.2. The Solution $b = 21$ Is A True Solution

Determine whether the given solutions are extraneous or true solutions for the following equations

Introduction

In mathematics, solving equations is a crucial aspect of problem-solving. However, it's not uncommon for solutions to be extraneous, meaning they don't satisfy the original equation. In this article, we'll explore two examples of equations and determine whether the given solutions are extraneous or true solutions.

Example 1: The solution $w = 8$ is an extraneous solution to the equation $\sqrt{2w} = -4$

Understanding the Equation

The given equation is $\sqrt{2w} = -4$. To determine whether the solution $w = 8$ is extraneous, we need to substitute $w = 8$ into the equation and check if it's true.

Substituting $w = 8$ into the Equation

Substituting $w = 8$ into the equation $\sqrt{2w} = -4$, we get:

$\sqrt{2(8)} = -4$

Simplifying the equation, we get:

$\sqrt{16} = -4$

Since $\sqrt{16} = 4$, not $-4$, the solution $w = 8$ is an extraneous solution to the equation $\sqrt{2w} = -4$.

Why is $w = 8$ an Extraneous Solution?

The solution $w = 8$ is an extraneous solution because it doesn't satisfy the original equation. When we substitute $w = 8$ into the equation, we get $\sqrt{16} = -4$, which is a contradiction. This means that $w = 8$ is not a valid solution to the equation $\sqrt{2w} = -4$.

Example 2: The solution $b = 21$ is a true solution to the equation $b^2 + 5b + 6 = 0$

Understanding the Equation

The given equation is $b^2 + 5b + 6 = 0$. To determine whether the solution $b = 21$ is a true solution, we need to substitute $b = 21$ into the equation and check if it's true.

Substituting $b = 21$ into the Equation

Substituting $b = 21$ into the equation $b^2 + 5b + 6 = 0$, we get:

$(21)^2 + 5(21) + 6 = 0$

Simplifying the equation, we get:

$441 + 105 + 6 = 0$

Combining like terms, we get:

$552 = 0$

Since $552 \neq 0$, the solution $b = 21$ is not a true solution to the equation $b^2 + 5b + 6 = 0$.

Why is $b = 21$ not a True Solution?

The solution $b = 21$ is not a true solution because it doesn't satisfy the original equation. When we substitute $b = 21$ into the equation, we get $552 = 0$, which is a contradiction. This means that $b = 21$ is not a valid solution to the equation $b^2 + 5b + 6 = 0$.

Conclusion

In conclusion, we've explored two examples of equations and determined whether the given solutions are extraneous or true solutions. In Example 1, we found that the solution $w = 8$ is an extraneous solution to the equation $\sqrt{2w} = -4$. In Example 2, we found that the solution $b = 21$ is not a true solution to the equation $b^2 + 5b + 6 = 0$. These examples demonstrate the importance of checking solutions to ensure they satisfy the original equation.

Tips for Checking Solutions

When checking solutions, it's essential to substitute the solution into the original equation and simplify the expression. If the resulting expression is a contradiction, then the solution is extraneous. If the resulting expression is true, then the solution is a true solution.

Common Mistakes to Avoid

When checking solutions, it's common to make mistakes such as:

• Not substituting the solution into the original equation
• Not simplifying the expression
• Not checking for contradictions

To avoid these mistakes, it's essential to carefully read the problem and follow the steps outlined above.

Final Thoughts

In conclusion, checking solutions is a crucial aspect of problem-solving in mathematics. By following the steps outlined above and avoiding common mistakes, you can ensure that your solutions are accurate and reliable. Remember, checking solutions is not just a necessary step, but also an opportunity to learn and improve your problem-solving skills.
Determine whether the given solutions are extraneous or true solutions for the following equations: Q&A

Introduction

In our previous article, we explored two examples of equations and determined whether the given solutions were extraneous or true solutions. In this article, we'll provide a Q&A section to help you better understand the concepts and apply them to your own problem-solving.

Q&A

Q: What is an extraneous solution?

A: An extraneous solution is a solution that doesn't satisfy the original equation. It's a solution that is not a valid solution to the equation.

Q: How do I determine whether a solution is extraneous or true?

A: To determine whether a solution is extraneous or true, you need to substitute the solution into the original equation and simplify the expression. If the resulting expression is a contradiction, then the solution is extraneous. If the resulting expression is true, then the solution is a true solution.

Q: What is a contradiction?

A: A contradiction is a statement that is false. In the context of solving equations, a contradiction occurs when the resulting expression is not true.

Q: Why is it important to check solutions?

A: Checking solutions is important because it ensures that the solution you found is a valid solution to the equation. If you don't check solutions, you may end up with an extraneous solution, which can lead to incorrect conclusions.

Q: What are some common mistakes to avoid when checking solutions?

A: Some common mistakes to avoid when checking solutions include:

• Not substituting the solution into the original equation
• Not simplifying the expression
• Not checking for contradictions

Q: How can I avoid making mistakes when checking solutions?

A: To avoid making mistakes when checking solutions, make sure to carefully read the problem and follow the steps outlined above. Take your time and double-check your work to ensure that you're getting the correct solution.

Q: What if I'm not sure whether a solution is extraneous or true?

A: If you're not sure whether a solution is extraneous or true, try substituting the solution into the original equation and simplifying the expression. If the resulting expression is a contradiction, then the solution is extraneous. If the resulting expression is true, then the solution is a true solution.

Q: Can I use a calculator to check solutions?

A: Yes, you can use a calculator to check solutions. However, make sure to double-check your work to ensure that you're getting the correct solution.

Q: How can I practice checking solutions?

A: To practice checking solutions, try working through some sample problems. You can find sample problems in your textbook or online. Make sure to carefully read the problem and follow the steps outlined above.

Conclusion

In conclusion, checking solutions is a crucial aspect of problem-solving in mathematics. By following the steps outlined above and avoiding common mistakes, you can ensure that your solutions are accurate and reliable. Remember, checking solutions is not just a necessary step, but also an opportunity to learn and improve your problem-solving skills.

Tips for Practicing Checking Solutions

• Start with simple problems and gradually move on to more complex ones.
• Make sure to carefully read the problem and follow the steps outlined above.
• Double-check your work to ensure that you're getting the correct solution.
• Use a calculator to check solutions, but make sure to double-check your work.
• Practice, practice, practice! The more you practice checking solutions, the more comfortable you'll become with the process.

Common Mistakes to Avoid

• Not substituting the solution into the original equation
• Not simplifying the expression
• Not checking for contradictions
• Using a calculator without double-checking your work
• Not practicing regularly

Final Thoughts

In conclusion, checking solutions is a crucial aspect of problem-solving in mathematics. By following the steps outlined above and avoiding common mistakes, you can ensure that your solutions are accurate and reliable. Remember, checking solutions is not just a necessary step, but also an opportunity to learn and improve your problem-solving skills.