Which Of The Following Is An Extraneous Solution Of $(45-3x)^{\frac{1}{2}} = X - 9$?A. X = − 12 X = -12 X = − 12 B. X = − 3 X = -3 X = − 3 C. X = 3 X = 3 X = 3 D. X = 12 X = 12 X = 12

Introduction

In mathematics, solving equations often involves finding the values of variables that satisfy the given equation. However, in some cases, the solutions obtained may not be valid or may not satisfy the original equation. These solutions are known as extraneous solutions. In this article, we will explore the concept of extraneous solutions and learn how to identify them in equations.

What are Extraneous Solutions?

Extraneous solutions are solutions that satisfy the equation but do not meet the conditions or constraints of the problem. These solutions can arise from various sources, such as:

• Invalid operations: When the equation involves operations that are not defined for certain values of the variable, such as taking the square root of a negative number.
• Domain restrictions: When the equation has domain restrictions, such as the square root function, which is only defined for non-negative values.
• Simplification errors: When simplifying the equation, we may introduce extraneous solutions that do not satisfy the original equation.

Identifying Extraneous Solutions

To identify extraneous solutions, we need to carefully analyze the equation and check for any conditions or constraints that may have been violated. Here are some common techniques to identify extraneous solutions:

• Check the domain: Verify that the solution satisfies the domain restrictions of the equation.
• Check the operations: Ensure that the operations involved in the equation are valid for the solution.
• Check the simplification: Verify that the simplification process did not introduce any extraneous solutions.

Solving the Given Equation

The given equation is:

$$(45-3x)^{\frac{1}{2}} = x - 9$$

To solve this equation, we can start by isolating the square root term:

$$(45-3x)^{\frac{1}{2}} = x - 9$$

$$45-3x = (x-9)^2$$

Expanding the right-hand side, we get:

$$45-3x = x^2 - 18x + 81$$

Rearranging the terms, we get:

$$x^2 - 15x + 36 = 0$$

This is a quadratic equation, which can be factored as:

$$(x-3)(x-12) = 0$$

Therefore, the solutions are $x = 3$ and $x = 12$.

Checking for Extraneous Solutions

To check for extraneous solutions, we need to verify that the solutions satisfy the domain restrictions of the equation. The square root function is only defined for non-negative values, so we need to check that the expression inside the square root is non-negative.

For $x = 3$, we have:

$$(45-3(3))^{\frac{1}{2}} = (45-9)^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6$$

Since $x = 3$ satisfies the domain restrictions, it is a valid solution.

For $x = 12$, we have:

$$(45-3(12))^{\frac{1}{2}} = (45-36)^{\frac{1}{2}} = 9^{\frac{1}{2}} = 3$$

Since $x = 12$ satisfies the domain restrictions, it is also a valid solution.

However, we need to check if $x = 3$ and $x = 12$ satisfy the original equation. Substituting $x = 3$ into the original equation, we get:

$$(45-3(3))^{\frac{1}{2}} = 3 - 9$$

$$6 = -6$$

This is a contradiction, so $x = 3$ is an extraneous solution.

Substituting $x = 12$ into the original equation, we get:

$$(45-3(12))^{\frac{1}{2}} = 12 - 9$$

$$3 = 3$$

This is a true statement, so $x = 12$ is a valid solution.

Conclusion

In conclusion, we have identified the extraneous solution of the given equation as $x = 3$. The other solution, $x = 12$, is a valid solution that satisfies the domain restrictions and the original equation.

Final Answer

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Q: What is an extraneous solution?

A: An extraneous solution is a solution that satisfies the equation but does not meet the conditions or constraints of the problem. These solutions can arise from various sources, such as invalid operations, domain restrictions, or simplification errors.

Q: How do I identify extraneous solutions?

A: To identify extraneous solutions, you need to carefully analyze the equation and check for any conditions or constraints that may have been violated. Here are some common techniques to identify extraneous solutions:

• Check the domain: Verify that the solution satisfies the domain restrictions of the equation.
• Check the operations: Ensure that the operations involved in the equation are valid for the solution.
• Check the simplification: Verify that the simplification process did not introduce any extraneous solutions.

Q: What are some common sources of extraneous solutions?

A: Some common sources of extraneous solutions include:

• Invalid operations: When the equation involves operations that are not defined for certain values of the variable, such as taking the square root of a negative number.
• Domain restrictions: When the equation has domain restrictions, such as the square root function, which is only defined for non-negative values.
• Simplification errors: When simplifying the equation, we may introduce extraneous solutions that do not satisfy the original equation.

Q: How do I check for extraneous solutions in a quadratic equation?

A: To check for extraneous solutions in a quadratic equation, you need to verify that the solutions satisfy the domain restrictions of the equation. For example, if the equation involves a square root, you need to check that the expression inside the square root is non-negative.

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

A: A valid solution is a solution that satisfies the equation and meets the conditions or constraints of the problem. An extraneous solution, on the other hand, is a solution that satisfies the equation but does not meet the conditions or constraints of the problem.

Q: Can an extraneous solution be a valid solution in a different context?

A: Yes, an extraneous solution can be a valid solution in a different context. For example, if the equation involves a square root, an extraneous solution may be valid in a different domain or with different operations.

Q: How do I avoid introducing extraneous solutions when simplifying equations?

A: To avoid introducing extraneous solutions when simplifying equations, you need to carefully analyze the equation and check for any conditions or constraints that may have been violated. Here are some common techniques to avoid introducing extraneous solutions:

• Check the domain: Verify that the solution satisfies the domain restrictions of the equation.
• Check the operations: Ensure that the operations involved in the equation are valid for the solution.
• Check the simplification: Verify that the simplification process did not introduce any extraneous solutions.

Q: What is the importance of identifying extraneous solutions?

A: Identifying extraneous solutions is important because it ensures that the solutions obtained are valid and meet the conditions or constraints of the problem. Extraneous solutions can lead to incorrect conclusions and affect the accuracy of the results.

Q: Can I use a calculator to check for extraneous solutions?

A: Yes, you can use a calculator to check for extraneous solutions. However, it is always a good idea to verify the results manually to ensure that the calculator is not introducing any errors.

Q: How do I know if a solution is an extraneous solution or not?

A: To determine if a solution is an extraneous solution or not, you need to carefully analyze the equation and check for any conditions or constraints that may have been violated. Here are some common techniques to determine if a solution is an extraneous solution or not:

• Check the domain: Verify that the solution satisfies the domain restrictions of the equation.
• Check the operations: Ensure that the operations involved in the equation are valid for the solution.
• Check the simplification: Verify that the simplification process did not introduce any extraneous solutions.

Conclusion

In conclusion, identifying extraneous solutions is an important step in solving equations. By carefully analyzing the equation and checking for any conditions or constraints that may have been violated, you can ensure that the solutions obtained are valid and meet the conditions or constraints of the problem.