Solve The Equation And Check For Extraneous Solutions.${ \frac{x+3}{x}+\frac{5}{x+8}+\frac{40}{x^2+8x}=0 }$ { x = [?] \}

Mar 7, 2025 by ADMIN 125 views

Solve the Equation and Check for Extraneous Solutions

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Introduction

In this article, we will be solving a complex equation involving fractions and then checking for any extraneous solutions. The given equation is $\frac{x+3}{x}+\frac{5}{x+8}+\frac{40}{x^2+8x}=0$ . Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Simplify the Equation

The first step in solving this equation is to simplify it by finding a common denominator for all the fractions. The common denominator for the fractions is $x(x+8)$ .

\frac{x+3}{x}+\frac{5}{x+8}+\frac{40}{x^2+8x} = \frac{(x+3)(x+8)}{x(x+8)} + \frac{5x}{x(x+8)} + \frac{40}{x(x+8)}

Step 2: Combine the Fractions

Now that we have a common denominator, we can combine the fractions into a single fraction.

\frac{(x+3)(x+8)}{x(x+8)} + \frac{5x}{x(x+8)} + \frac{40}{x(x+8)} = \frac{(x+3)(x+8) + 5x + 40}{x(x+8)}

Step 3: Expand the Numerator

Next, we need to expand the numerator of the fraction.

(x+3)(x+8) + 5x + 40 = x^2 + 11x + 24 + 5x + 40

Step 4: Combine Like Terms

Now, we can combine like terms in the numerator.

x^2 + 11x + 24 + 5x + 40 = x^2 + 16x + 64

Step 5: Rewrite the Equation

Now that we have simplified the numerator, we can rewrite the equation.

\frac{x^2 + 16x + 64}{x(x+8)} = 0

Step 6: Factor the Numerator

The numerator of the fraction can be factored as follows:

x^2 + 16x + 64 = (x+8)^2

Step 7: Rewrite the Equation

Now that we have factored the numerator, we can rewrite the equation.

\frac{(x+8)^2}{x(x+8)} = 0

Step 8: Cancel Out Common Factors

We can cancel out the common factors in the numerator and denominator.

\frac{(x+8)^2}{x(x+8)} = \frac{x+8}{x}

Step 9: Solve for x

Now that we have simplified the equation, we can solve for $x$ .

\frac{x+8}{x} = 0

Step 10: Check for Extraneous Solutions

To check for extraneous solutions, we need to plug the value of $x$ back into the original equation and check if it is true.

x = -8

Plugging $x = -8$ back into the original equation, we get:

\frac{-8+3}{-8}+\frac{5}{-8+8}+\frac{40}{(-8)^2+8(-8)} = \frac{-5}{-8}+\frac{5}{0}+\frac{40}{64-64}

Conclusion

The final answer is $\boxed{-8}$ . However, we need to check if this solution is extraneous. Plugging $x = -8$ back into the original equation, we get an undefined value, which means that $x = -8$ is an extraneous solution.

Final Answer

The final answer is $\boxed{no solution}$ .

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Introduction

In our previous article, we solved the equation $\frac{x+3}{x}+\frac{5}{x+8}+\frac{40}{x^2+8x}=0$ and found that the solution was $x = -8$ . However, we also found that this solution was extraneous. In this article, we will answer some frequently asked questions about solving the equation and checking for extraneous solutions.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not actually a solution to the equation. In other words, it is a solution that makes the equation true, but only because of a mistake or a misunderstanding.

Q: How do I know if a solution is extraneous?

A: To check if a solution is extraneous, you need to plug it back into the original equation and check if it is true. If the solution makes the equation true, but only because of a mistake or a misunderstanding, then it is an extraneous solution.

Q: Why do I need to check for extraneous solutions?

A: You need to check for extraneous solutions because they can be misleading. If you find a solution and don't check for extraneous solutions, you may think that it is the only solution, but it may actually be an extraneous solution.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. You can also use algebraic methods, such as factoring or simplifying the equation, to check for extraneous solutions.

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

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

Not checking for extraneous solutions
Not plugging the solution back into the original equation
Not using algebraic methods to check for extraneous solutions
Making mistakes when simplifying or factoring the equation

Q: How can I avoid extraneous solutions?

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

Check for extraneous solutions
Plug the solution back into the original equation
Use algebraic methods to check for extraneous solutions
Double-check your work to make sure that you haven't made any mistakes

Q: What are some tips for solving equations and checking for extraneous solutions?

A: Here are some tips for solving equations and checking for extraneous solutions:

Read the problem carefully and make sure you understand what is being asked
Use algebraic methods to simplify and solve the equation
Check for extraneous solutions by plugging the solution back into the original equation
Use factoring or simplifying the equation to check for extraneous solutions
Double-check your work to make sure that you haven't made any mistakes

Conclusion

Solving equations and checking for extraneous solutions can be challenging, but with practice and patience, you can become proficient. Remember to be careful and methodical when solving equations, and make sure to check for extraneous solutions. By following these tips and being mindful of common mistakes, you can avoid extraneous solutions and find the correct solution to the equation.

Final Answer

The final answer is $\boxed{no solution}$ .