What Is The Extraneous Solution To The Equation?$\frac{4x+13}{x-1}=\frac{x^2+4x+4}{x-1}$A. There Are No Extraneous Solutions. B. $x = -3$ C. $x = 0$ D. $x = -1$

What is the Extraneous Solution to the Equation?

Understanding the Concept of Extraneous Solutions

In mathematics, particularly in algebra, an extraneous solution is a solution that appears to be valid but is actually incorrect. This concept is crucial when solving equations, especially those involving fractions or rational expressions. In this article, we will explore the concept of extraneous solutions and how to identify them.

What is an Extraneous Solution?

An extraneous solution is a solution that satisfies an equation but is not a valid solution in the context of the problem. This can occur when there are restrictions on the values of the variables, such as division by zero or taking the square root of a negative number. Extraneous solutions can also arise when simplifying expressions or canceling out common factors.

The Given Equation

The given equation is $\frac{4x+13}{x-1}=\frac{x^2+4x+4}{x-1}$. To solve this equation, we need to first eliminate the fractions by multiplying both sides by the common denominator, which is $x-1$. However, we must be cautious when doing so, as this may introduce extraneous solutions.

Solving the Equation

To solve the equation, we multiply both sides by $x-1$:

$\frac{4x+13}{x-1}=\frac{x^2+4x+4}{x-1}$

$(4x+13) = (x^2+4x+4)$

Now, we can simplify the equation by expanding and combining like terms:

$4x+13 = x^2+4x+4$

Subtracting $4x$ from both sides gives:

$13 = x^2+4$

Subtracting $4$ from both sides gives:

$9 = x^2$

Taking the square root of both sides gives:

$\pm 3 = x$

Identifying the Extraneous Solution

Now that we have found the solutions to the equation, we need to check if any of them are extraneous. To do this, we need to check if the solutions satisfy the original equation and if they are valid in the context of the problem.

Let's check the solution $x = 3$:

$\frac{4(3)+13}{3-1}=\frac{(3)^2+4(3)+4}{3-1}$

$\frac{12+13}{2}=\frac{9+12+4}{2}$

$\frac{25}{2}=\frac{25}{2}$

This solution satisfies the original equation, so it is a valid solution.

Now, let's check the solution $x = -3$:

$\frac{4(-3)+13}{-3-1}=\frac{(-3)^2+4(-3)+4}{-3-1}$

$\frac{-12+13}{-4}=\frac{9-12+4}{-4}$

$\frac{1}{-4}=\frac{1}{-4}$

This solution also satisfies the original equation, so it is a valid solution.

However, we need to check if these solutions are valid in the context of the problem. In this case, the original equation involves division by $x-1$, which is not defined when $x = 1$. Therefore, we need to check if $x = 1$ is a solution to the equation.

Let's check if $x = 1$ is a solution:

$\frac{4(1)+13}{1-1}=\frac{(1)^2+4(1)+4}{1-1}$

This expression is undefined, as division by zero is not allowed. Therefore, $x = 1$ is not a solution to the equation.

However, we notice that $x = 1$ is not among the solutions we found earlier. This means that the solutions we found are not extraneous, and they are valid solutions to the equation.

Conclusion

In conclusion, the extraneous solution to the equation $\frac{4x+13}{x-1}=\frac{x^2+4x+4}{x-1}$ is not among the solutions we found. The solutions $x = 3$ and $x = -3$ are valid solutions to the equation, and they satisfy the original equation. Therefore, the correct answer is:

A. There are no extraneous solutions.

Note: The other options are incorrect, as $x = 0$ and $x = -1$ are not solutions to the equation.
Q&A: Extraneous Solutions

What is an Extraneous Solution?

An extraneous solution is a solution that appears to be valid but is actually incorrect. This can occur when there are restrictions on the values of the variables, such as division by zero or taking the square root of a negative number.

Q: How do I identify an Extraneous Solution?

A: To identify an extraneous solution, you need to check if the solution satisfies the original equation and if it is valid in the context of the problem. You can do this by plugging the solution back into the original equation and checking if it is true.

Q: What are some common ways that Extraneous Solutions can occur?

A: Extraneous solutions can occur when:

• You divide by zero
• You take the square root of a negative number
• You cancel out common factors that are not valid
• You simplify expressions in a way that introduces extraneous solutions

Q: How can I avoid Extraneous Solutions?

A: To avoid extraneous solutions, you need to be careful when solving equations and simplifying expressions. Make sure to:

• Check your work carefully
• Avoid dividing by zero
• Be careful when canceling out common factors
• Check if the solutions you find are valid in the context of the problem

Q: Can Extraneous Solutions be eliminated?

A: Yes, extraneous solutions can be eliminated by checking if the solutions you find satisfy the original equation and if they are valid in the context of the problem.

Q: What is the difference between a valid solution and an extraneous solution?

A: A valid solution is a solution that satisfies the original equation and is valid in the context of the problem. An extraneous solution is a solution that appears to be valid but is actually incorrect.

Q: Can I use the same method to solve equations with extraneous solutions?

A: Yes, you can use the same method to solve equations with extraneous solutions. However, you need to be careful and check your work carefully to avoid introducing extraneous solutions.

Q: How can I determine if a solution is extraneous?

A: To determine if a solution is extraneous, you need to check if the solution satisfies the original equation and if it is valid in the context of the problem. You can do this by plugging the solution back into the original equation and checking if it is true.

Q: What are some common mistakes that can lead to Extraneous Solutions?

A: Some common mistakes that can lead to extraneous solutions include:

• Dividing by zero
• Taking the square root of a negative number
• Canceling out common factors that are not valid
• Simplifying expressions in a way that introduces extraneous solutions

Q: Can I use technology to help me identify Extraneous Solutions?

A: Yes, you can use technology to help you identify extraneous solutions. For example, you can use a graphing calculator to graph the equation and check if the solutions you find are valid.

Q: How can I practice identifying Extraneous Solutions?

A: You can practice identifying extraneous solutions by working on problems that involve equations with extraneous solutions. Make sure to check your work carefully and use technology to help you if needed.

Conclusion

In conclusion, extraneous solutions are a common problem that can occur when solving equations. By understanding what an extraneous solution is and how to identify it, you can avoid making mistakes and find the correct solutions to equations. Remember to check your work carefully and use technology to help you if needed.