Find The Real Roots Of The Equation ${ \frac{18}{x 4}+\frac{1}{x 2}=4 }$

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Introduction

In this article, we will delve into finding the real roots of the given equation $\frac{18}{x^4}+\frac{1}{x^2}=4$. This equation involves a quartic term and can be solved using various mathematical techniques. Our goal is to find the real values of $x$ that satisfy the equation.

Understanding the Equation

The given equation is $\frac{18}{x^4}+\frac{1}{x^2}=4$. To simplify the equation, we can multiply both sides by $x^4$ to eliminate the fractions. This gives us $18 + x^2 = 4x^4$.

Rearranging the Equation

We can rearrange the equation to get $4x^4 - x^2 - 18 = 0$. This is a quartic equation in terms of $x^2$, and we can use various techniques to solve it.

Using the Quadratic Formula

Since the equation is in terms of $x^2$, we can substitute $y = x^2$ to get $4y^2 - y - 18 = 0$. This is a quadratic equation in terms of $y$, and we can use the quadratic formula to solve it.

The quadratic formula is given by $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 4$, $b = -1$, and $c = -18$. Plugging these values into the quadratic formula, we get:

$$y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-18)}}{2(4)}$$

Simplifying the expression, we get:

$$y = \frac{1 \pm \sqrt{1 + 288}}{8}$$

$$y = \frac{1 \pm \sqrt{289}}{8}$$

$$y = \frac{1 \pm 17}{8}$$

This gives us two possible values for $y$: $y = \frac{1 + 17}{8} = 2.25$ and $y = \frac{1 - 17}{8} = -2$.

Substituting Back $x^2$

We can substitute back $x^2 = y$ to get $x^2 = 2.25$ and $x^2 = -2$. However, since $x^2$ cannot be negative, we discard the solution $x^2 = -2$.

Finding the Real Roots

We are left with $x^2 = 2.25$. Taking the square root of both sides, we get $x = \pm \sqrt{2.25}$. Simplifying the expression, we get $x = \pm 1.5$.

Conclusion

In this article, we found the real roots of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$. The real roots are $x = 1.5$ and $x = -1.5$. These values satisfy the equation and can be used to solve various mathematical problems.

Discussion

The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ is a quartic equation in terms of $x^2$. We used the quadratic formula to solve the equation and found the real roots. The real roots are $x = 1.5$ and $x = -1.5$. These values can be used to solve various mathematical problems.

Applications

The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ has various applications in mathematics and physics. For example, it can be used to model the behavior of a physical system with a quartic potential. The real roots of the equation can be used to find the equilibrium points of the system.

Future Work

In future work, we can explore other techniques for solving the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$. For example, we can use numerical methods to find the roots of the equation. We can also explore the properties of the equation and its roots.

References

• [1] "Quadratic Formula" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadraticformula.html
• [2] "Quartic Equation" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/QuarticEquation.html

Keywords

• Quartic equation
• Quadratic formula
• Real roots
• Mathematical techniques
• Physical systems
• Equilibrium points
  

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Q&A: Finding the Real Roots of the Equation

Q: What is the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ is a quartic equation in terms of $x^2$. It can be simplified to $4x^4 - x^2 - 18 = 0$.

Q: How do I solve the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: To solve the equation, we can use the quadratic formula. We can substitute $y = x^2$ to get $4y^2 - y - 18 = 0$. Then, we can use the quadratic formula to find the values of $y$. Finally, we can substitute back $x^2 = y$ to find the values of $x$.

Q: What are the real roots of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: The real roots of the equation are $x = 1.5$ and $x = -1.5$. These values satisfy the equation and can be used to solve various mathematical problems.

Q: How do I use the real roots of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: The real roots of the equation can be used to find the equilibrium points of a physical system with a quartic potential. They can also be used to solve various mathematical problems.

Q: What are some applications of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ has various applications in mathematics and physics. For example, it can be used to model the behavior of a physical system with a quartic potential.

Q: Can I use numerical methods to find the roots of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: Yes, you can use numerical methods to find the roots of the equation. However, the quadratic formula is a more efficient and accurate method for solving the equation.

Q: What are some properties of the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$?

A: The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ is a quartic equation in terms of $x^2$. It has two real roots, $x = 1.5$ and $x = -1.5$. The equation can be used to model the behavior of a physical system with a quartic potential.

Q: Can I use the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ to solve other mathematical problems?

A: Yes, you can use the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ to solve other mathematical problems. For example, you can use the equation to find the equilibrium points of a physical system with a quartic potential.

Conclusion

In this article, we have discussed the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ and its real roots. We have also answered some common questions about the equation and its applications. The equation has various applications in mathematics and physics, and its real roots can be used to solve various mathematical problems.

Discussion

The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ is a quartic equation in terms of $x^2$. It has two real roots, $x = 1.5$ and $x = -1.5$. The equation can be used to model the behavior of a physical system with a quartic potential. The real roots of the equation can be used to find the equilibrium points of the system.

Applications

The equation $\frac{18}{x^4}+\frac{1}{x^2}=4$ has various applications in mathematics and physics. For example, it can be used to model the behavior of a physical system with a quartic potential. The real roots of the equation can be used to find the equilibrium points of the system.

Future Work

In future work, we can explore other techniques for solving the equation $\frac{18}{x^4}+\frac{1}{x^2}=4$. For example, we can use numerical methods to find the roots of the equation. We can also explore the properties of the equation and its roots.

References

• [1] "Quadratic Formula" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadraticformula.html
• [2] "Quartic Equation" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/QuarticEquation.html

Keywords

• Quartic equation
• Quadratic formula
• Real roots
• Mathematical techniques
• Physical systems
• Equilibrium points