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

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

  • Square roots: When an equation involves square roots, the solution may not be valid if the expression inside the square root is negative.
  • Fractional exponents: When an equation involves fractional exponents, the solution may not be valid if the base is negative.
  • Absolute values: When an equation involves absolute values, the solution may not be valid if the expression inside the absolute value is negative.

Identifying Extraneous Solutions

To identify extraneous solutions, we need to check if the solution satisfies the original equation and meets the conditions or constraints of the equation. Here are some steps to follow:

  1. Check the domain: Check if the solution is within the domain of the equation. For example, if the equation involves a square root, check if the expression inside the square root is non-negative.
  2. Check the range: Check if the solution is within the range of the equation. For example, if the equation involves a logarithm, check if the argument is positive.
  3. Check the constraints: Check if the solution meets the constraints of the equation. For example, if the equation involves an absolute value, check if the expression inside the absolute value is non-negative.

Example: Solving for Extraneous Solutions

Let's consider the equation:

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

We need to find the extraneous solutions of this equation. To do this, we will follow the steps outlined above.

Step 1: Check the Domain

The equation involves a square root, so we need to check if the expression inside the square root is non-negative.

(45−3x)≥0(45-3x) \geq 0

Simplifying the inequality, we get:

15−x≥015-x \geq 0

Solving for x, we get:

x≤15x \leq 15

Step 2: Check the Range

The equation does not involve any logarithms, so we do not need to check the range.

Step 3: Check the Constraints

The equation involves an absolute value, but it is not necessary to check the constraints in this case.

Step 4: Solve the Equation

Now that we have checked the domain, range, and constraints, we can solve the equation.

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

Squaring both sides of the equation, we get:

45−3x=x2−18x+8145-3x=x^2-18x+81

Simplifying the equation, we get:

x2−15x+36=0x^2-15x+36=0

Factoring the quadratic equation, we get:

(x−3)(x−12)=0(x-3)(x-12)=0

Solving for x, we get:

x=3 or x=12x=3 \text{ or } x=12

Step 5: Check for Extraneous Solutions

Now that we have found the solutions, we need to check if they are extraneous. We will check if the solutions satisfy the original equation and meet the conditions or constraints of the equation.

For x = 3:

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

Simplifying the equation, we get:

(36)12=−6(36)^{\frac{1}{2}}=-6

This is not true, so x = 3 is an extraneous solution.

For x = 12:

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

Simplifying the equation, we get:

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

This is true, so x = 12 is not an extraneous solution.

Conclusion

In this example, we found that x = 3 is an extraneous solution of the equation $(45-3x)^{\frac{1}{2}}=x-9$. This is because x = 3 does not satisfy the original equation and meets the conditions or constraints of the equation.

Answer

The correct answer is:

  • A. x=−12x=-12 is not an extraneous solution.
  • B. x=−3x=-3 is not an extraneous solution.
  • C. x=3x=3 is an extraneous solution.
  • D. x=12x=12 is not an extraneous solution.

Final Thoughts

Q: What is an extraneous solution?

A: An extraneous solution is a value of a variable that satisfies an equation, but does not meet the conditions or constraints of the equation.

Q: Why do extraneous solutions occur?

A: Extraneous solutions can occur due to various reasons, such as:

  • Square roots: When an equation involves square roots, the solution may not be valid if the expression inside the square root is negative.
  • Fractional exponents: When an equation involves fractional exponents, the solution may not be valid if the base is negative.
  • Absolute values: When an equation involves absolute values, the solution may not be valid if the expression inside the absolute value is negative.

Q: How do I identify extraneous solutions?

A: To identify extraneous solutions, follow these steps:

  1. Check the domain: Check if the solution is within the domain of the equation. For example, if the equation involves a square root, check if the expression inside the square root is non-negative.
  2. Check the range: Check if the solution is within the range of the equation. For example, if the equation involves a logarithm, check if the argument is positive.
  3. Check the constraints: Check if the solution meets the constraints of the equation. For example, if the equation involves an absolute value, check if the expression inside the absolute value is non-negative.

Q: What are some common types of extraneous solutions?

A: Some common types of extraneous solutions include:

  • Negative square roots: When an equation involves a square root, the solution may not be valid if the expression inside the square root is negative.
  • Negative fractional exponents: When an equation involves a fractional exponent, the solution may not be valid if the base is negative.
  • Negative absolute values: When an equation involves an absolute value, the solution may not be valid if the expression inside the absolute value is negative.

Q: How do I eliminate extraneous solutions?

A: To eliminate extraneous solutions, follow these steps:

  1. Check the original equation: Check if the solution satisfies the original equation.
  2. Check the conditions or constraints: Check if the solution meets the conditions or constraints of the equation.
  3. Eliminate the extraneous solution: If the solution does not satisfy the original equation or meets the conditions or constraints, eliminate it as an extraneous solution.

Q: What are some real-world applications of extraneous solutions?

A: Extraneous solutions have various real-world applications, such as:

  • Engineering: Extraneous solutions can occur in engineering problems, such as designing a bridge or a building. In these cases, the solution must meet the conditions or constraints of the problem.
  • Physics: Extraneous solutions can occur in physics problems, such as calculating the trajectory of a projectile. In these cases, the solution must meet the conditions or constraints of the problem.
  • Computer Science: Extraneous solutions can occur in computer science problems, such as solving a system of linear equations. In these cases, the solution must meet the conditions or constraints of the problem.

Q: How do I avoid extraneous solutions in the future?

A: To avoid extraneous solutions in the future, follow these tips:

  • Read the problem carefully: Read the problem carefully and understand the conditions or constraints of the problem.
  • Check the domain and range: Check if the solution is within the domain and range of the equation.
  • Check the conditions or constraints: Check if the solution meets the conditions or constraints of the equation.
  • Eliminate extraneous solutions: Eliminate any solutions that do not meet the conditions or constraints of the equation.

Conclusion

In conclusion, extraneous solutions are values of variables that satisfy an equation, but do not meet the conditions or constraints of the equation. To identify extraneous solutions, follow the steps outlined above. By understanding extraneous solutions, you can avoid making mistakes and ensure that your solutions are valid.