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I am making a math project for my school. We can make it on any topic, but should involve some college level math. I have chosen 'Cardano's method' as my topic. I will be showing the method to solve a general cubic equation$$ax^3+bx^2+cx+d=0$$ using this method and also show how an example with given coefficients a, b, c and d. The problem is I must also be able to solve a real life problem (even if it's hypothetical) using whatever math I am using. At first I considered the real gas equation: $$(P+\frac{an^2}{V^2})(V-nb)=nRT$$ On knowing all the parameters except volume, we get a cubic equation is V. I first tried to plus in some real values for oxygen gas, but the equation I received contained very huge numbers and was difficult to solve. I tried considering hypothetical values of Vander Waal's constant a and b and yet it was pretty difficult to solve using cardano's method. Is there a simple real life problem which requires us to find the roots of a cubic equation and gives simple answers?

Deepani Agarwal
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    Why did you choose Cardano's method? It is often written that the explicit formulas for roots of cubic and quartic polynomials are useless in practical settings, and I think that's correct. These formulas are related to interesting topics in pure math, but they're essentially irrelevant to applications of math in any direct way. Historically, special cases of Cardano's formula are what led mathematicians to begin paying attention to complex numbers (see the Veritasium video https://www.youtube.com/watch?v=cUzklzVXJwo), and complex numbers are very useful, but not Cardano's formula itself. – KCd Jan 13 '23 at 05:59
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    Note I am not saying cubic polynomials never show up in some applied problem, but if one does and there is not an obvious root of it (if the equation were $x^3 - a = 0$ then we'd just use $\sqrt[3]{a}$), you just rely on the computer to solve it. Nobody pulls out Cardano's formula as a step towards solving the problem. – KCd Jan 13 '23 at 06:05
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    Perhaps, a better application is to illustrate Newton's method for solving polynomial equations. Show them that no matter how complicated the equation is, the algorithm runs very fast. And in "real life" this is usually what is done with equations, as the above people have commented on. – Nicolas Bourbaki Jan 13 '23 at 06:26
  • @KCd I think you are right, Cardano's method is not the best idea fo rmy project. Is there something similar that you suggest which can make a good project? – Deepani Agarwal Jan 13 '23 at 07:23
  • Newton's method was suggested already as an alternative, and it seems like a good idea. The 3Blue1Brown video https://www.youtube.com/watch?v=-RdOwhmqP5s has a nice discussion of unexpected mathematical aspects of Newton's method, but note what he says during 3:45-5:03 about solving cubics and quartics. A visualization of the 2-dimensional Newton's method is here: https://www.youtube.com/watch?v=fuHl4bgCQTI. – KCd Jan 13 '23 at 08:09

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I won't give you an explicit question, but I suggest you think of problems like this: A right cylindrical cone has base radius $R$ and height $H$. A right circular cylinder of base radius $r$ and height $h$ is inscribed in the cone. Find its dimensions if its volume is a given fraction $c$ times the volume of the cone.

You can think of other geometric problems with cubes inscribed in spheres, etc.

P.S. I'm not sure I consider this college level math; certainly the basics of complex numbers and DeMoivre's formula shouldn't be.

Ted Shifrin
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If you want to solve for $V$ a cubic equation of state, you will face major problems (just as you already noticed).

It is much better to solve it for $Z$, the compressibility factor $Z=\frac{P V}{nRT}$. Then the equation to be solved becomes $$Z^3- \left(1+\frac{b P}{R T}\right)Z^2+\frac{a P}{R^2 T^2}Z-\frac{a b P^2}{R^3 T^3}=0$$

For a real gas, except for very high pressures, the solution will be rather close to $Z=1$ and Newton method will converge as a charm. Do not use Cardano's method.

Now, jus kidding you a little bit, my answer to your question

Is there a simple real life problem which requires us to find the roots of a cubic equation and gives simple answers?

will be : What have I been doing over the last fifty years ?.

If this is of interest for you, search for my papers about cubic equations of state.

Edit

For a first approximation and then as a initial value for Newton method, you can use (for a gas phase) $$Z_0=1+\left(b-\frac{a}{R T} \right)\frac P{R T}$$

Claude Leibovici
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