Quadratic equations using complex math but with no imaginary roots

Question

Many years ago when learning complex maths we used complex maths as an example in the quadratic equation to find real roots. My nephew is struggling to deal with complex maths as his teacher is teaching it as an academic exercise. I am tutoring himn and my background is engineering so I prefer real examples and using complex maths to solve the quadratic equation for real roots seems to me to be a useful example.

Can anyone recall any real roots that require the use of complex maths to calculate? Or did I imagine it?

Answer 1

Per request, I am expanding my comment into an answer. Given a quadratic equation, $ax^2+bx+c=0$, the quadratic formula $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ will only involve complex numbers exactly when the discriminant $b^2-4ac<0$. In this case the two roots come in a conjugate pair and there are no real solutions. However, when $b^2-4ac \geq 0$, there is no need to invoke complex numbers. Solutions may involve square roots, but they are of positive numbers.

However, if we wish to solve a cubic equation, the corresponding cubic formula will involve complex numbers even when all of the roots are real. Let's look at Cardano's formula for the depressed cubic, $x^3+px+q=0$. (depressed=no quadratic term) In this case $$ x = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$$ (See https://www.encyclopediaofmath.org/index.php/Cardano_formula)

When the cubic discriminant $D=-27q^2-4p^3$ is positive, the roots are real and distinct, but the computation involves complex numbers in the intermediary steps. (Look at the square roots.) If I recall correctly, this formula is the first occurrence of the complex numbers.

As an addition, complex numbers often show in the context of some kind of system which oscillates or rotates. It might be worthwhile for your nephew to look at examples of the form $i^n$ and $ \frac{ i^n + (-i)^n}{2}$ for $n=0,1,2,\ldots$ It would be good practice and illustrates an important point.

Answer 2

(Quarter loaf and too long for a comment)

1. In integration of rational functions, using method of partial fractions, you can often have individual terms that have denominators which are complex-rooted quadratics. So this is a touchpoint, maybe. Yet the integral itself is still a real valued function. (Whether definite or indefinite integral.)

2. Obviously many aspects of EE use complex numbers to show phase angle and the like. You know ELI the ICE man. 3 phase y-delta all that. A basic EE course of 1-2 semesters is standard for all engineers (even mechanical, etc.) Of course there are more complex (pun intended) aspects of EE courses that use complex algebra or even calculus (signal processing, spectrum analysis stuff).

3. Most engineers (of whatever slant, but definitely mechanical and EE) have to take a control systems class. This ends up having at its core the solution of the second order constant coefficient diffyQ (with forcing function). The solution of the homo part of this problem ends up using complex numbers in the underdamped case. (or sin, cos...but it is very useful to kind of "see" this problem both in terms of sin/cos or in terms of complex exponentials.)

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Note: Maybe this helps your neph to feel better. However, I still suspect it is going to be a hard slog for him. The real value of the complex applications (mostly) comes from the very first chapter of a complex analysis text (where you revisit algebra two complex algebra, do the Argand diagram, solve for the square root of i, etc.).

Yes, there are applications of the analysis part of complex analysis also, but they will not be needed for anything in the stereotypical undergrad engineering curriculum...are more for grad students. And even here, I bet your instructor is not going to teach things in a "how to solve integrals" manner, but with a lot of emphasis on proofs.

Probably the kid would have been better off in a complex variables (engineering emphasis) class than this theory ball-buster. But even that would have been overkill for what he really needs.

My best advice is to just tell him to go whole hog and do things how the teacher wants. Fatalistically. He can always try to construct meaning from it later on his own. But if he struggles too much with that now, it will derail him. Tell him to "get proofy". Camp out on that bastard's office doorstep for extra instruction help. Ask why "five times" until he gets it. Be PERSISTENT. And do lots of homework. Maybe find easier, transitional books if the text is also a ball buster. Read the text AHEAD of lecture and work the problems (at least try to) ahead of lecture. [Standard advice for mastering any subject, but especially needed to pull out all the study skills stops, since the class is more difficult than normal.]

Good luck to the young man. He will need it.