Solve For \[$ X \$\]:$\[ \begin{array}{c} 5x - \frac{16}{x} = -16 \\ x = [?] \end{array} \\]

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Introduction

In this article, we will delve into solving a complex equation involving fractions and variables. The equation given is 5x - 16/x = -16, and we need to find the value of x that satisfies this equation. This type of equation is commonly encountered in algebra and requires a step-by-step approach to solve.

Understanding the Equation

The given equation is 5x - 16/x = -16. To solve for x, we need to isolate x on one side of the equation. However, the presence of the fraction 16/x makes it challenging to isolate x directly. We will need to employ algebraic techniques to simplify the equation and solve for x.

Multiplying Both Sides by x

One way to simplify the equation is to multiply both sides by x. This will eliminate the fraction and make it easier to work with the equation. Multiplying both sides by x gives us:

5x^2 - 16 = -16x

Rearranging the Equation

Now that we have eliminated the fraction, we can rearrange the equation to make it easier to solve. We can add 16x to both sides of the equation to get:

5x^2 + 16x = 0

Factoring the Equation

The equation 5x^2 + 16x = 0 can be factored using the distributive property. We can factor out x from both terms to get:

x(5x + 16) = 0

Solving for x

Now that we have factored the equation, we can solve for x. We can set each factor equal to zero and solve for x. Setting x = 0 gives us:

x = 0

Setting 5x + 16 = 0 gives us:

5x = -16

x = -16/5

Conclusion

In this article, we have solved the equation 5x - 16/x = -16 using algebraic techniques. We multiplied both sides by x to eliminate the fraction, rearranged the equation to make it easier to solve, and factored the equation to find the values of x that satisfy the equation. The solutions to the equation are x = 0 and x = -16/5.

Final Answer

The final answer to the equation 5x - 16/x = -16 is x = 0 and x = -16/5.

Discussion

The equation 5x - 16/x = -16 is a complex equation that requires a step-by-step approach to solve. The presence of the fraction 16/x makes it challenging to isolate x directly, but by multiplying both sides by x, we can eliminate the fraction and make it easier to work with the equation. The solutions to the equation are x = 0 and x = -16/5.

Related Topics

  • Solving equations with fractions
  • Factoring quadratic equations
  • Algebraic techniques for solving equations

References

  • [1] Algebra: A Comprehensive Introduction, by Michael Artin
  • [2] Calculus: Early Transcendentals, by James Stewart
  • [3] Precalculus: Mathematics for Calculus, by James Stewart

Keywords

  • Solving equations with fractions
  • Factoring quadratic equations
  • Algebraic techniques for solving equations
  • Complex equations
  • Algebra
  • Mathematics

Introduction

In our previous article, we solved the equation 5x - 16/x = -16 using algebraic techniques. In this article, we will answer some frequently asked questions related to solving this equation.

Q: What is the first step in solving the equation 5x - 16/x = -16?

A: The first step in solving the equation 5x - 16/x = -16 is to multiply both sides by x. This will eliminate the fraction and make it easier to work with the equation.

Q: Why do we need to multiply both sides by x?

A: We need to multiply both sides by x to eliminate the fraction 16/x. This will make it easier to isolate x and solve the equation.

Q: What is the next step after multiplying both sides by x?

A: After multiplying both sides by x, we need to rearrange the equation to make it easier to solve. We can add 16x to both sides of the equation to get 5x^2 + 16x = 0.

Q: How do we solve the equation 5x^2 + 16x = 0?

A: We can solve the equation 5x^2 + 16x = 0 by factoring the equation. We can factor out x from both terms to get x(5x + 16) = 0.

Q: What are the solutions to the equation 5x - 16/x = -16?

A: The solutions to the equation 5x - 16/x = -16 are x = 0 and x = -16/5.

Q: Why are there two solutions to the equation 5x - 16/x = -16?

A: There are two solutions to the equation 5x - 16/x = -16 because the equation is a quadratic equation. Quadratic equations can have two solutions, which are the values of x that satisfy the equation.

Q: What is the significance of the solutions x = 0 and x = -16/5?

A: The solutions x = 0 and x = -16/5 are significant because they represent the values of x that satisfy the equation 5x - 16/x = -16. These values can be used to solve problems and make predictions in various fields such as science, engineering, and economics.

Q: How can I apply the solutions to the equation 5x - 16/x = -16 in real-life situations?

A: The solutions to the equation 5x - 16/x = -16 can be applied in various real-life situations such as solving problems in physics, engineering, and economics. For example, the solution x = 0 can be used to represent a situation where there is no movement or change, while the solution x = -16/5 can be used to represent a situation where there is a negative movement or change.

Q: What are some common mistakes to avoid when solving the equation 5x - 16/x = -16?

A: Some common mistakes to avoid when solving the equation 5x - 16/x = -16 include:

  • Not multiplying both sides by x to eliminate the fraction
  • Not rearranging the equation to make it easier to solve
  • Not factoring the equation correctly
  • Not checking the solutions to make sure they satisfy the equation

Q: How can I practice solving equations like 5x - 16/x = -16?

A: You can practice solving equations like 5x - 16/x = -16 by working on similar problems and exercises. You can also use online resources and tools to help you practice and improve your skills.

Q: What are some additional resources that can help me learn more about solving equations like 5x - 16/x = -16?

A: Some additional resources that can help you learn more about solving equations like 5x - 16/x = -16 include:

  • Algebra textbooks and workbooks
  • Online tutorials and videos
  • Practice problems and exercises
  • Online communities and forums

Conclusion

In this article, we have answered some frequently asked questions related to solving the equation 5x - 16/x = -16. We have covered topics such as the first step in solving the equation, why we need to multiply both sides by x, and how to solve the equation. We have also discussed the significance of the solutions and how to apply them in real-life situations.