What is an example of real application of cubic equations?

One example ...

The curves used in Postscript (including Postscript fonts), and in most drawing and graphics programs (like Adobe Illustrator, Powerpoint, etc.), are cubic Bézier curves.

The outlines of the characters you are reading right now might be strings of cubic Bézier curves. The prefered font for this page is "Georgia", which uses (TrueType) quadratic curves. But, if you don't have the Georgia font installed, Times Roman will be displayed instead, and that uses cubic curves. The seemingly circular dot above the "i" is actually four cubic curves.

Similarly, whenever you read a PDF document, or almost any printed document, you are (often) looking at strings of cubic curves.

Graphics packages like OpenGL and Direct3D also use cubic curves heavily. So the curves you see in games and other 3D programs are often cubics.

Whenever you do any calculations with cubic Bezier curves, you are using cubic equations. For example, if you calculate points on the curves so that you can draw them on a computer screen or a printer.

If you want to intersect one of these curves with a straight line, you will have to solve a cubic equation. This happens (for example) when you "clip" the curve to some rectangular boundary.

In short, cubic Bézier curves are everywhere. You can start reading about all of this here.

If you want to dig a lot deeper, you could search for the word "cubic" in this bibliography.

Eigenvalues and Eigenvectors are an essential tool in the theory of matrix computation. The case of $3\times3$ matrices is particularly important as it relates to geometric transforms of our good old 3D world.

The Eigenvalues of a $3\times3$ matrix are found from this appetizing equation:

$$x^3-Tr(A)x^2+\frac12[Tr^2(A)-Tr(A^2)]x-det(A)=0$$

As a "real world" example, consider fitting a plane to a point cloud in 3D space (for instance to model the facets of an object digitized with a 3D scanner). This leads to an Eigen problem of this kind (or similarly Singular Value Decomposition).

The drag of airplanes is essentially the coefficient of drag of the airframe (complete) times velocity cubed.

So for a $10\%$ increase in speed, it requires $1.1^3$ more horse power ($33.1\%$ more horsepower) providing that:

1. the additional horsepower adds no additional weight (not likely)
2. the additional horsepower does not change the airframe's outline or shape in any way (except perhaps for a propeller’s blade profile, planform, or blade count) (also, not likely)

That is why recent advances in airplane design have focused on the two primary sources of drag and a third secondary (Which are LINEAR):

1. Airframe and wetted area drag. (parasitic drag)
2. Coefficient of drag for the lift produced by the wing. (Induced Drag)
3. Drag produced lifting more or less weight. (eventually becomes Induced Drag)

The joke is that a aerodynamicist designing a plane would sell his/her grandmother for a few drag points, or counts at the third decimal place of the coefficient of drag, i.e. $C(f)= 0.26x$.