# An engineer's approach to the quadratic formula

• DaveE
In summary, the conversation discusses a paper from a former EE instructor that highlights the difference between memorizing a mathematical result and truly understanding it. The instructor emphasizes the importance of simplifying solutions in engineering for practical use and the value of approximation in design. The conversation also mentions the availability of free papers on the website and the instructor's approach to teaching at CalTech.
DaveE
Gold Member
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

Nik_2213, MatinSAR, .Scott and 4 others

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.

Nik_2213

## What is an engineer's approach to the quadratic formula?

An engineer's approach to the quadratic formula often emphasizes practical application and approximation rather than purely theoretical derivation. Engineers might focus on numerical methods, simplifications, and real-world applicability when solving quadratic equations.

## How does an engineer simplify the quadratic formula for practical use?

Engineers may use approximations or iterative methods to simplify the quadratic formula. For example, they might employ numerical algorithms like the Newton-Raphson method or use simplified forms of the formula when certain terms are negligible in practical scenarios.

## Why do engineers prefer numerical methods over exact solutions for quadratic equations?

Engineers often deal with real-world problems where exact solutions are either unnecessary or impractical due to measurement errors or complex models. Numerical methods provide sufficiently accurate solutions more efficiently, especially when dealing with large systems or real-time applications.

## Can you provide an example of a real-world application where an engineer would use the quadratic formula?

One common application is in the analysis of projectile motion. Engineers use the quadratic formula to determine the time of flight, range, and maximum height of a projectile by solving the equations of motion under the influence of gravity.

## What tools do engineers use to solve quadratic equations?

Engineers often use software tools such as MATLAB, Python (with libraries like NumPy), or specialized engineering calculators to solve quadratic equations. These tools allow for quick and accurate computations, including handling complex or large systems of equations.

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