Solving the Equation x^4 - x^2 - 1 0 Using Complex Roots: A Comprehensive Guide

When dealing with polynomial equations, particularly those that do not have real roots, the realm of complex numbers becomes essential. The equation x^4 - x^2 - 1 0 is a perfect example of such a scenario. In this article, we will explore various methods to solve this equation, including the use of complex roots, the quadratic formula, and the polar form of complex numbers.

Understanding the Equation

The equation x^4 - x^2 - 1 0 can be analyzed by substituting y x^2. This simplifies the equation to a more manageable form:

y^2 - y - 1 0

Using the Quadratic Formula

To find the values of y, we apply the quadratic formula:

y frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 1, b -1, and c -1. Substituting these values, we get:

y frac{1 pm sqrt{(-1)^2 - 4 cdot 1 cdot (-1)}}{2 cdot 1}

frac{1 pm sqrt{1 4}}{2}

frac{1 pm sqrt{5}}{2}

Since y x^2, we have:

x^2 frac{1 pm sqrt{5}}{2}

Complex Roots

When x^2 frac{1 pm sqrt{5}}{2}, the solutions for x involve complex roots. We express these roots as:

x pm sqrt{frac{1 pm sqrt{5}}{2}}

Alternative Solution Using Complex Numbers

We can also solve this equation by expressing it in a different form. Consider the equation:

x^4 - x^2 - 1 0

By multiplying both sides by x^2 1, we get:

x^6 - 1 0

Or,

x^6 e^{2npi i}

where n is an integer. The solutions for x can then be given by:

x e^{frac{2npi i}{6}} cos frac{2npi}{6} i sin frac{2npi}{6}

Evaluated for n 0, 1, 2, 3, 4, 5, we obtain the following roots:

Thus, the solutions for x are:

x frac{sqrt{3}i}{2}, i, frac{-sqrt{3}i}{2}, frac{-sqrt{3}-i}{2}, -i, frac{sqrt{3}-i}{2}

Conclusion

The equation x^4 - x^2 - 1 0 can be solved using various methods, including the quadratic formula, substitution, and the polar form of complex numbers. These methods reveal the intricacies of solving polynomial equations with complex roots, highlighting the importance of understanding the behavior of these equations in the complex plane.