I have been learning the possible solutions for this type of equations but I have no idea when and how it can be used in real life. Do we have any problematic example for a real life application of quartic equations? How finding the roots of them help us in real life for real life problems that we may face?


  • $\begingroup$ A lot of things in physics. Too many to mention, frankly. $\endgroup$ Dec 9, 2019 at 14:06
  • $\begingroup$ @DonThousand I really would like to know about some of them. $\endgroup$
    – Mathrix
    Dec 9, 2019 at 14:09
  • 1
    $\begingroup$ If you want to find the points of intersection of two ellipses (described by their equations), you have to solve a quartic. $\endgroup$
    – awkward
    Dec 9, 2019 at 14:22
  • $\begingroup$ @awkward thanks, but can this be considered as a real life problem? $\endgroup$
    – Mathrix
    Dec 9, 2019 at 14:23
  • 2
    $\begingroup$ Define "real life". What sort of applications are you after? For example, finance is full of hard maths. $\endgroup$
    – user1729
    Dec 9, 2019 at 18:42

3 Answers 3


This is from Wikipedia

Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus. It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are examples of other geometric problems whose solution involves solving a quartic equation.

In computer-aided manufacturing, the torus is a shape that is commonly associated with the endmill cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the z-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.[8]

A quartic equation arises also in the process of solving the crossed ladders problem, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.

In optics, Alhazen's problem is "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to a quartic equation.[9][10][11]

Finding the distance of closest approach of two ellipses involves solving a quartic equation.

The eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix.

The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending.

Intersections between spheres, cylinders, or other quadrics can be found using quartic equations.


One real example is the curve of deflection of a beam supported on the two ends and loaded by a continuous load. Another example is the curve of slope of a beam loaded by a triangular load. You can read any textbook on strength of materials or mechanics of materials if you are not from mechanical or civil engineering.


Waves, for example electromagnetic waves, have both a frequency, the number of wave cycles per unit time, and a wave-number, the number of wave cycles per unit length. (The wave-number is the reciprocal of the more familiar wave-length). The relationship between frequency and wave-number may depend upon the wave-number (or equivalently upon the frequency) giving rise to the phenomenon known as dispersion (which is responsible for the the spreading of colors as light passes through a prism). The relationship between wave-number and frequency is called the dispersion relationship. A model for a common scenario for the propagation of waves is what is known as a transmission line. The dispersion relationship for a transmission line is quartic.

$$4\beta^4 + 4a_0\beta^2 = 4a_2\beta^2\omega^2 + a_1^2\omega^2$$


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