Which Solution To The Equation 3 A + 2 + 2 A = 4 A − 4 A 2 − 4 \frac{3}{a+2}+\frac{2}{a}=\frac{4a-4}{a^2-4} A + 2 3 ​ + A 2 ​ = A 2 − 4 4 A − 4 ​ Is Extraneous?A. A = − 2 A = -2 A = − 2 B. A = − 2 A = -2 A = − 2 And A = 4 A = 4 A = 4 C. Neither A = − 2 A = -2 A = − 2 Nor A = 4 A = 4 A = 4 D. A = 4 A = 4 A = 4

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Introduction

When solving equations involving fractions, it's essential to consider the possibility of extraneous solutions. An extraneous solution is a value that satisfies the equation but is not a valid solution due to the restrictions imposed by the original equation. In this article, we will explore the concept of extraneous solutions and apply it to the given equation $\frac{3}{a+2}+\frac{2}{a}=\frac{4a-4}{a^2-4}$.

Understanding Extraneous Solutions

Extraneous solutions can arise when the original equation contains fractions or other restrictions that limit the possible values of the variable. In the case of the given equation, we need to consider the restrictions imposed by the denominators of the fractions.

Restrictions Imposed by the Denominators

The denominators of the fractions in the equation are $a+2$ and $a$. To avoid division by zero, we must ensure that these denominators are not equal to zero. Therefore, we have the following restrictions:

• $a+2 \neq 0$
• $a \neq 0$

Solving the Equation

To solve the equation, we can start by finding a common denominator for the fractions on the left-hand side. The common denominator is $(a+2)a$. Multiplying both sides of the equation by the common denominator, we get:

$$3a + 4(a+2) = (4a-4)a$$

Expanding and simplifying the equation, we get:

$$3a + 4a + 8 = 4a^2 - 4a$$

Combine like terms:

$$7a + 8 = 4a^2 - 4a$$

Rearrange the equation to get a quadratic equation in standard form:

$$4a^2 - 11a - 8 = 0$$

Factoring the Quadratic Equation

We can factor the quadratic equation as follows:

$$(4a+8)(a-1) = 0$$

Solving for $a$

To find the values of $a$ that satisfy the equation, we can set each factor equal to zero and solve for $a$:

$$4a+8 = 0 \Rightarrow a = -2$$

$$a-1 = 0 \Rightarrow a = 1$$

Checking for Extraneous Solutions

We need to check if the values of $a$ we found are valid solutions to the original equation. We can do this by plugging each value of $a$ back into the original equation and checking if the equation holds true.

For $a = -2$, we get:

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

Simplifying the equation, we get:

$$\frac{3}{0}+\frac{2}{-2}=\frac{-12-4}{4-4}$$

This equation is undefined, so $a = -2$ is an extraneous solution.

For $a = 1$, we get:

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

Simplifying the equation, we get:

$$\frac{3}{3}+\frac{2}{1}=\frac{0}{-3}$$

This equation is true, so $a = 1$ is a valid solution.

Conclusion

In conclusion, the value of $a$ that is extraneous to the equation $\frac{3}{a+2}+\frac{2}{a}=\frac{4a-4}{a^2-4}$ is $a = -2$. Therefore, the correct answer is:

A. $a = -2$

Note that the other options are incorrect because $a = 4$ is not a solution to the equation, and neither $a = -2$ nor $a = 4$ is the correct answer.

Q: What is an extraneous solution?

A: An extraneous solution is a value that satisfies an equation but is not a valid solution due to the restrictions imposed by the original equation.

Q: Why do extraneous solutions occur?

A: Extraneous solutions can occur when the original equation contains fractions or other restrictions that limit the possible values of the variable.

Q: How do I identify extraneous solutions?

A: To identify extraneous solutions, you need to check if the values you found satisfy the original equation and if they are valid solutions. You can do this by plugging the values back into the original equation and checking if the equation holds true.

Q: What are some common restrictions that can lead to extraneous solutions?

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

• Division by zero
• Square roots of negative numbers
• Logarithms of non-positive numbers
• Fractions with denominators that are equal to zero

Q: How do I avoid extraneous solutions when solving equations?

A: To avoid extraneous solutions, you need to:

• Check the restrictions imposed by the original equation
• Plug the values back into the original equation to check if they are valid solutions
• Consider the domain of the equation and the possible values of the variable

Q: Can extraneous solutions be eliminated by simply ignoring them?

A: No, extraneous solutions cannot be eliminated by simply ignoring them. They are still valid solutions to the equation, but they are not valid in the context of the original problem.

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

A: To determine if a solution is extraneous or not, you need to:

• Check if the solution satisfies the original equation
• Check if the solution is valid in the context of the original problem
• Consider the domain of the equation and the possible values of the variable

Q: Can extraneous solutions be avoided by using a different method to solve the equation?

A: Yes, extraneous solutions can be avoided by using a different method to solve the equation. For example, you can use algebraic manipulations or numerical methods to solve the equation.

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

A: To know if a solution is valid or not, you need to:

• Check if the solution satisfies the original equation
• Check if the solution is valid in the context of the original problem
• Consider the domain of the equation and the possible values of the variable

Q: Can extraneous solutions be eliminated by using a calculator or computer?

A: No, extraneous solutions cannot be eliminated by using a calculator or computer. They are still valid solutions to the equation, but they are not valid in the context of the original problem.

Q: How do I handle multiple extraneous solutions?

A: If you have multiple extraneous solutions, you need to:

• Check each solution individually to determine if it is valid or not
• Consider the domain of the equation and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the equation and eliminate extraneous solutions.

Q: Can extraneous solutions be avoided by using a specific type of equation?

A: Yes, extraneous solutions can be avoided by using a specific type of equation, such as a linear equation or a quadratic equation. However, even with these types of equations, extraneous solutions can still occur if the equation is not properly defined.

Q: How do I know if an equation has extraneous solutions or not?

A: To know if an equation has extraneous solutions or not, you need to:

• Check the restrictions imposed by the original equation
• Plug the values back into the original equation to check if they are valid solutions
• Consider the domain of the equation and the possible values of the variable

Q: Can extraneous solutions be eliminated by using a specific type of function?

A: Yes, extraneous solutions can be eliminated by using a specific type of function, such as a linear function or a quadratic function. However, even with these types of functions, extraneous solutions can still occur if the function is not properly defined.

Q: How do I handle extraneous solutions in a system of equations?

A: If you have a system of equations and you find an extraneous solution, you need to:

• Check each equation individually to determine if the solution is valid or not
• Consider the domain of each equation and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the system of equations and eliminate extraneous solutions.

Q: Can extraneous solutions be avoided by using a specific type of method?

A: Yes, extraneous solutions can be avoided by using a specific type of method, such as the substitution method or the elimination method. However, even with these methods, extraneous solutions can still occur if the equation is not properly defined.

Q: How do I know if a solution is valid or not in a system of equations?

A: To know if a solution is valid or not in a system of equations, you need to:

• Check each equation individually to determine if the solution is valid or not
• Consider the domain of each equation and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the system of equations and eliminate extraneous solutions.

Q: Can extraneous solutions be eliminated by using a specific type of software?

A: No, extraneous solutions cannot be eliminated by using a specific type of software. They are still valid solutions to the equation, but they are not valid in the context of the original problem.

Q: How do I handle extraneous solutions in a graphing problem?

A: If you have a graphing problem and you find an extraneous solution, you need to:

• Check each point individually to determine if it is valid or not
• Consider the domain of the graph and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the problem and eliminate extraneous solutions.

Q: Can extraneous solutions be avoided by using a specific type of graph?

A: Yes, extraneous solutions can be avoided by using a specific type of graph, such as a linear graph or a quadratic graph. However, even with these types of graphs, extraneous solutions can still occur if the graph is not properly defined.

Q: How do I know if a solution is valid or not in a graphing problem?

A: To know if a solution is valid or not in a graphing problem, you need to:

• Check each point individually to determine if it is valid or not
• Consider the domain of the graph and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the problem and eliminate extraneous solutions.

Q: Can extraneous solutions be eliminated by using a specific type of calculator?

A: No, extraneous solutions cannot be eliminated by using a specific type of calculator. They are still valid solutions to the equation, but they are not valid in the context of the original problem.

Q: How do I handle extraneous solutions in a numerical problem?

A: If you have a numerical problem and you find an extraneous solution, you need to:

• Check each value individually to determine if it is valid or not
• Consider the domain of the problem and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the problem and eliminate extraneous solutions.

Q: Can extraneous solutions be avoided by using a specific type of numerical method?

A: Yes, extraneous solutions can be avoided by using a specific type of numerical method, such as the Newton-Raphson method or the bisection method. However, even with these methods, extraneous solutions can still occur if the equation is not properly defined.

Q: How do I know if a solution is valid or not in a numerical problem?

A: To know if a solution is valid or not in a numerical problem, you need to:

• Check each value individually to determine if it is valid or not
• Consider the domain of the problem and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the problem and eliminate extraneous solutions.

Q: Can extraneous solutions be eliminated by using a specific type of software?

A: No, extraneous solutions cannot be eliminated by using a specific type of software. They are still valid solutions to the equation, but they are not valid in the context of the original problem.

Q: How do I handle extraneous solutions in a word problem?

A: If you have a word problem and you find an extraneous solution, you need to:

• Check each value individually to determine if it is valid or not
• Consider the domain of the problem and the possible values of the variable
• Use algebraic manipulations or numerical methods to solve the problem and eliminate extraneous solutions.

Q: Can extraneous solutions be avoided by using a specific type of word problem?

A: Yes, extraneous solutions can be avoided by using a specific type of word problem, such as a linear word problem or a quadratic word problem. However, even with these types of word problems, extraneous solutions can still occur if the problem is not properly defined.

Q: How do I know if a solution is valid or not in a word problem?

A: To know if a solution is valid or not in a word problem, you need to:

• Check each value individually to determine if it is valid or not
• Consider the domain