Solve For { X $}$ In The Equation:${ X^4 - 81 = 0 }$

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Introduction

Understanding the Equation The given equation is a fourth-degree polynomial equation, which can be solved using various methods. In this article, we will focus on solving the equation $x^4 - 81 = 0$ to find the values of $x$. This equation can be solved using factorization, which is a powerful method for solving polynomial equations.

Step 1: Factorize the Equation

The equation $x^4 - 81 = 0$ can be factorized as follows:

${ x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9) = 0 }$

This factorization is based on the difference of squares formula, which states that $a^2 - b^2 = (a - b)(a + b)$.

Step 2: Solve for $x^2 - 9 = 0$

We can now solve for $x^2 - 9 = 0$ by adding 9 to both sides of the equation:

${ x^2 - 9 + 9 = 0 + 9 }$ ${ x^2 = 9 }$

Step 3: Solve for $x^2 + 9 = 0$

We can now solve for $x^2 + 9 = 0$ by subtracting 9 from both sides of the equation:

${ x^2 + 9 - 9 = 0 - 9 }$ ${ x^2 = -9 }$

Step 4: Solve for $x$ in $x^2 = 9$

We can now solve for $x$ in $x^2 = 9$ by taking the square root of both sides of the equation:

${ x^2 = 9 }$ ${ x = \pm \sqrt{9} }$ ${ x = \pm 3 }$

Step 5: Solve for $x$ in $x^2 = -9$

We can now solve for $x$ in $x^2 = -9$ by taking the square root of both sides of the equation:

${ x^2 = -9 }$ ${ x = \pm \sqrt{-9} }$ ${ x = \pm 3i }$

Conclusion

In this article, we have solved the equation $x^4 - 81 = 0$ using factorization. We have found that the solutions to this equation are $x = \pm 3$ and $x = \pm 3i$. These solutions can be verified by substituting them back into the original equation.

Applications of the Equation

The equation $x^4 - 81 = 0$ has several applications in mathematics and physics. For example, it can be used to model the motion of a particle in a quadratic potential. It can also be used to solve problems involving the roots of a polynomial equation.

Real-World Examples

The equation $x^4 - 81 = 0$ can be used to model real-world problems such as:

• The motion of a particle in a quadratic potential
• The roots of a polynomial equation
• The behavior of a system with a quadratic potential

Future Research Directions

There are several future research directions that can be explored using the equation $x^4 - 81 = 0$. For example:

• Using the equation to model more complex systems
• Exploring the applications of the equation in physics and engineering
• Developing new methods for solving polynomial equations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Acknowledgments

The author would like to acknowledge the support of the [Name of Institution] in completing this research. The author would also like to thank [Name of Person] for their helpful comments and suggestions.

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Introduction

In our previous article, we solved the equation $x^4 - 81 = 0$ using factorization. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the difference between $x^4 - 81 = 0$ and $x^2 - 9 = 0$?

A: The equation $x^4 - 81 = 0$ is a fourth-degree polynomial equation, while the equation $x^2 - 9 = 0$ is a quadratic equation. The difference between the two equations is that the former has a higher degree and more complex solutions.

Q: How do I factorize the equation $x^4 - 81 = 0$?

A: To factorize the equation $x^4 - 81 = 0$, you can use the difference of squares formula, which states that $a^2 - b^2 = (a - b)(a + b)$. In this case, you can factorize the equation as $(x^2 - 9)(x^2 + 9) = 0$.

Q: What are the solutions to the equation $x^4 - 81 = 0$?

A: The solutions to the equation $x^4 - 81 = 0$ are $x = \pm 3$ and $x = \pm 3i$. These solutions can be verified by substituting them back into the original equation.

Q: Can I use other methods to solve the equation $x^4 - 81 = 0$?

A: Yes, you can use other methods to solve the equation $x^4 - 81 = 0$, such as the quadratic formula or numerical methods. However, factorization is a powerful method for solving polynomial equations and is often the most efficient method.

Q: What are the applications of the equation $x^4 - 81 = 0$?

A: The equation $x^4 - 81 = 0$ has several applications in mathematics and physics, such as modeling the motion of a particle in a quadratic potential or solving problems involving the roots of a polynomial equation.

Q: Can I use the equation $x^4 - 81 = 0$ to model real-world problems?

A: Yes, you can use the equation $x^4 - 81 = 0$ to model real-world problems, such as the motion of a particle in a quadratic potential or the behavior of a system with a quadratic potential.

Q: What are some future research directions for the equation $x^4 - 81 = 0$?

A: Some future research directions for the equation $x^4 - 81 = 0$ include using the equation to model more complex systems, exploring the applications of the equation in physics and engineering, and developing new methods for solving polynomial equations.

Q: Where can I find more information about solving the equation $x^4 - 81 = 0$?

A: You can find more information about solving the equation $x^4 - 81 = 0$ in textbooks on algebra and calculus, as well as online resources such as Khan Academy and Wolfram Alpha.

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $x^4 - 81 = 0$. We hope that this article has been helpful in providing a better understanding of this equation and its applications.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Acknowledgments

The author would like to acknowledge the support of the [Name of Institution] in completing this research. The author would also like to thank [Name of Person] for their helpful comments and suggestions.