An engineer's approach to the quadratic formula

I stumbled across this article from decades past written by the best EE instructor I ever had. I thought it might be of some passing interest to others in highlighting the difference between memorizing a mathematical result vs. truly understanding it. The essence of engineering, in effect. We all know the quadratic formula, but do you ever really think about what it means, how it works in practice?

https://authors.library.caltech.edu/63245/1/00683365.pdf

BTW, this site has some really good papers, most of which you can read for free.

Possibly because that 'original' version could be derived / solved without melting us hapless students' brains ?

Also, yes, it was expected you'd hand-calculate it, possibly with aid of slide-rule, four-figure logs, six if fussy.
"TEN significant figures ? You doing orbits, tide-charts or something ??"

Akin to way the generic 'Standard Deviation' formula does not suit use in a computer algorithm, requiring loops through stored data. Collecting the 'needful' as data entered is much faster and more accurate...

Nik_2213 said:

Possibly because that 'original' version could be derived / solved without melting us hapless students' brains ?

Yes, it's easy to remember a solution by 'completing the square'. His point is, both here and more generally in engineering, that you aren't done with your derivation until you get the result into a simple, practical, form. Simple forms increase understanding and allow you to make good decisions in design efforts. Things like simplification with good approximations for example. This isn't often taught to university level students they're stuck with their high school "hapless students' brain version".

He used to say "Engineering is the art of approximation", which I think is true. It should be explicitly taught, as he did at CalTech. He has several more complex versions like this in Analog EE analysis. I like this one because everyone uses the quadratic formula and everyone thinks they know all about it from high school.