Solving Quartic Equations: Real and Complex Roots

A quartic equation is a polynomial equation of the fourth degree. While it might seem simple at first glance, the solutions can be quite complex, especially when they involve both real and complex roots. Let's delve into a specific quartic equation and explore how to solve it for both its real and complex roots.

Understanding the Quartic Equation

The fourth-degree polynomial equation given is:

x4 - x2 - 4x 12

This equation can be rewritten as:

x4 - x2 - 4x - 12 0

Solution of the Quartic Equation

While there exists a general solution to quartic equations, it is not always straightforward to apply. The roots of this particular quartic equation are two real roots and two complex roots. Specifically, the real roots are approximately -1.68692 and 2.26159, and the complex roots are approximately -0.287336 - 1.75009i and -0.287336 1.75009i. The complex roots are complex conjugates of each other, which is a common occurrence when solving quartic equations.

Graphical Representation

The graphical plot of the function shows that the equation has a wonky curve with only two solutions. The solutions are approximately at x -1.686920796652746 and x 2.261592104989187. The curve can be visualized in the range of x from -3.5 to 3.5, as shown below.

Closed Form Solution

For quartic equations, a closed form solution is available, but it is generally quite complex and involves nested radicals. For the equation x4 - x2 - 4x - 12 0, the closed form solutions can be derived using the general method for solving quartic equations, which involves several algebraic steps. However, the exact closed form expression is not provided in this article due to its complexity.

Applications and Real-World Relevance

Quartic equations have applications in various fields, including physics, engineering, and economics. For example, in physics, the motion of a particle under certain forces can be described by a quartic equation. In engineering, the design of certain mechanical systems can involve solving quartic equations to ensure stability and functionality. In economics, models of market dynamics often incorporate polynomial equations, some of which can be quartics.

Conclusion

In conclusion, quartic equations like x4 - x2 - 4x - 12 0 can have a mixture of real and complex roots. Understanding how to solve these equations is crucial in various practical applications. The real roots of this equation are approximately -1.68692 and 2.26159, while the complex roots are approximately -0.287336 - 1.75009i and -0.287336 1.75009i. The graphical representation and closed form solution provide valuable insights into the nature of these roots and the function's behavior.