Select All Equations For Which -64 Is A Solution.A. X = 8 \sqrt{x} = 8 X ​ = 8 B. X = − 8 \sqrt{x} = -8 X ​ = − 8 C. X 3 = 4 \sqrt[3]{x} = 4 3 X ​ = 4 D. X 3 = − 4 \sqrt[3]{x} = -4 3 X ​ = − 4 E. − X = 8 -\sqrt{x} = 8 − X ​ = 8 F. − X = 8 \sqrt{-x} = 8 − X ​ = 8

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Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. However, not all equations have real solutions, and some may have complex or imaginary solutions. In this article, we will focus on selecting equations for which -64 is a solution. We will analyze each given equation and determine whether it satisfies the condition.

Understanding the Concept of Solutions

Before we dive into the equations, let's understand what a solution is. A solution to an equation is a value that makes the equation true. In other words, when we substitute the solution into the equation, the equation holds true. For example, if we have the equation x + 5 = 10, the solution is x = 5, because when we substitute x = 5 into the equation, it becomes 5 + 5 = 10, which is true.

Analyzing the Equations

Now, let's analyze each equation and determine whether -64 is a solution.

A. x=8\sqrt{x} = 8

To determine whether -64 is a solution, we need to substitute -64 into the equation and see if it holds true.

−64=(−4)2=4≠8\sqrt{-64} = \sqrt{(-4)^2} = 4 \neq 8

Since −64≠8\sqrt{-64} \neq 8, -64 is not a solution to this equation.

B. x=−8\sqrt{x} = -8

Similarly, we need to substitute -64 into the equation and see if it holds true.

−64=(−4)2=4≠−8\sqrt{-64} = \sqrt{(-4)^2} = 4 \neq -8

Since −64≠−8\sqrt{-64} \neq -8, -64 is not a solution to this equation.

C. x3=4\sqrt[3]{x} = 4

To determine whether -64 is a solution, we need to substitute -64 into the equation and see if it holds true.

−643=(−4)33=−4≠4\sqrt[3]{-64} = \sqrt[3]{(-4)^3} = -4 \neq 4

Since −643≠4\sqrt[3]{-64} \neq 4, -64 is not a solution to this equation.

D. x3=−4\sqrt[3]{x} = -4

Similarly, we need to substitute -64 into the equation and see if it holds true.

−643=(−4)33=−4=−4\sqrt[3]{-64} = \sqrt[3]{(-4)^3} = -4 = -4

Since −643=−4\sqrt[3]{-64} = -4, -64 is a solution to this equation.

E. −x=8-\sqrt{x} = 8

To determine whether -64 is a solution, we need to substitute -64 into the equation and see if it holds true.

−−64=−(−4)2=−4≠8-\sqrt{-64} = -\sqrt{(-4)^2} = -4 \neq 8

Since −−64≠8-\sqrt{-64} \neq 8, -64 is not a solution to this equation.

F. −x=8\sqrt{-x} = 8

Similarly, we need to substitute -64 into the equation and see if it holds true.

−(−64)=64=8\sqrt{-(-64)} = \sqrt{64} = 8

Since −(−64)=8\sqrt{-(-64)} = 8, -64 is a solution to this equation.

Conclusion

In conclusion, we have analyzed each equation and determined whether -64 is a solution. The equations that have -64 as a solution are:

  • x3=−4\sqrt[3]{x} = -4
  • −x=8\sqrt{-x} = 8

These equations satisfy the condition, and -64 is indeed a solution to these equations.

Final Thoughts

Solving equations is a crucial concept in mathematics, and understanding the concept of solutions is essential. By analyzing each equation and determining whether -64 is a solution, we have gained a deeper understanding of the subject. We hope that this article has provided valuable insights and helped readers develop their problem-solving skills.

References

Additional Resources

Note: The references and additional resources provided are for informational purposes only and are not an endorsement of any particular website or resource.

Introduction

In our previous article, we analyzed several equations and determined whether -64 is a solution. However, we understand that readers may still have questions and concerns about the topic. In this article, we will address some of the most frequently asked questions related to selecting equations with solutions.

Q&A

Q: What is a solution to an equation?

A: A solution to an equation is a value that makes the equation true. In other words, when we substitute the solution into the equation, the equation holds true.

Q: How do I determine whether -64 is a solution to an equation?

A: To determine whether -64 is a solution to an equation, we need to substitute -64 into the equation and see if it holds true. If the equation is true when we substitute -64, then -64 is a solution to the equation.

Q: What is the difference between a real solution and a complex solution?

A: A real solution is a value that makes the equation true and is a real number. A complex solution is a value that makes the equation true and is a complex number.

Q: Can -64 be a solution to an equation if it is a complex number?

A: Yes, -64 can be a solution to an equation if it is a complex number. In fact, -64 is a complex number because it is the square of a negative number.

Q: How do I know whether an equation has a real solution or a complex solution?

A: To determine whether an equation has a real solution or a complex solution, we need to analyze the equation and see if it can be solved using real numbers or complex numbers.

Q: What is the significance of selecting equations with solutions?

A: Selecting equations with solutions is an important concept in mathematics because it helps us understand the properties of equations and how to solve them. It also helps us develop problem-solving skills and critical thinking.

Q: Can I use a calculator to determine whether -64 is a solution to an equation?

A: Yes, you can use a calculator to determine whether -64 is a solution to an equation. However, it is also important to understand the underlying mathematics and be able to solve the equation manually.

Q: How do I choose the correct equation to solve?

A: To choose the correct equation to solve, we need to analyze the problem and determine which equation is relevant to the situation. We also need to consider the constraints and limitations of the problem.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions related to selecting equations with solutions. We hope that this article has provided valuable insights and helped readers develop their problem-solving skills.

Final Thoughts

Selecting equations with solutions is an important concept in mathematics that requires critical thinking and problem-solving skills. By understanding the properties of equations and how to solve them, we can develop a deeper understanding of the subject and improve our ability to solve problems.

References

Additional Resources

Note: The references and additional resources provided are for informational purposes only and are not an endorsement of any particular website or resource.