Mathematician Finds Easier Way to Solve Quadratic Equations

A mathematician at Carnegie Mellon University has developed an easier way to solve quadratic equations.

The mathematician hopes this method will help students avoid memorizing obtuse formulas.

His secret is in generalizing two roots together instead of keeping them as separate values.

A mathematician has derived an easier way to solve quadratic equation problems, according to MIT's Technology Review. Quadratic equations are polynomials that include an x², and teachers use them to teach students to find two solutions at once. The new process, developed by Dr. Po-Shen Loh at Carnegie Mellon University, goes around traditional methods like completing the square and turns finding roots into a simpler thing involving fewer steps that are also more intuitive. Here's Dr. Loh's explainer video.

Quadratic equations fall into an interesting donut hole in education. Students learn them beginning in algebra or pre-algebra classes, but they’re spoonfed examples that work out very easily and with whole integer solutions. The same thing happens with the Pythagorean theorem, where in school, most examples end up solving out to Pythagorean triples, the small set of integer values that work cleanly into the Pythagorean theorem.

Quadratic equations are polynomials, meaning strings of math terms. An expression like “x + 4” is a polynomial. They can have one or many variables in any combination, and the magnitude of them is decided by what power the variables are taken to. So x + 4 is an expression describing a straight line, but (x + 4)² is a curve. Since a line crosses just once through any particular latitude or longitude, its solution is just one value. If you have x², that means two root values, in a shape like a circle or arc that makes two crossings.

Dr. Loh’s method, which he also shared in detail on his website, uses the idea of the two roots of every quadratic equation to make a simpler way to derive those roots. He realized he could describe the two roots of a quadratic equation this way: Combined, they average out to a certain value, then there’s a value z that shows any additional unknown value. Instead of searching for two separate, different values, we’re searching for two identical values to begin with. This simplifies the arithmetic part of multiplying the formula out.

Dr. Loh believes students can learn this method more intuitively, partly because there’s not a special, separate formula required. If students can remember some simple generalizations about roots, they can decide where to go next. It’s still complicated, but it’s less complicated, especially if Dr. Loh is right that this will smooth students’s understanding of how quadratic equations work and how they fit into math. Understanding them is key to the beginning ideas of precalculus, for example.

Outside of classroom-ready examples, the quadratic method isn't simple. Real examples and applications are messy, with ugly roots made of decimals or irrational numbers. As a student, it's hard to know you've found the right answer. Dr. Loh’s new method is for real life, but he hopes it will also help students feel they understand the quadratic formula better at the same time. Many math students struggle to move across the gulf in understanding between simple classroom examples and applying ideas themselves, and Dr. Loh wants to build them a better bridge.

"... you just divide both sides by a."

Of course.

But this does introduce an additional step.

Furthermore, the video is actually doing the general solution without realizing that it is doing so.

Note that he mentions equidistance of solutions by suggesting that the sum of the two solutions must be 8. He concludes that the solutions are (4+u) and (4-u).

He could have just as easily said that (-b/2a) is the average of the two solutions, which not surprisingly is easily calculated as, ... wait for it ..., 4.

The rest of the calculation is u = sqrt(b²-4ac)/2a or, since a=1,
u = sqrt(b²-4c)/2 = sqrt(64-48)/2 = sqrt(16)/2 = 2.

I just don't see the advantage, especially when using such simple coefficients.

Furthermore, I notice what I think is a slight weakness in his derivation. His initial assumption is that the roots are (4+u) and (4-u). He is considering u to be either positive for both roots or negative for both roots, but not both. Then he multiplies the two roots, simplifies, and recognizes that there are two solutions for u.

It works in this case, but I would be very reluctant to carry out such a calculation without some consideration of which root to use. It is possible to introduce impossible solutions with such steps. I think we have all seen the calculations that prove that 2=1 or some such contradiction by dividing by zero without recognizing it.

I’m a Mechanical Engineer, and minored in Math (MANY years ago...). I memorized multiplication tables in 3rd grade, the quadratic equation in Junior High, and many other things along the way (including some manipulations in Calculus).

THERE IS NOTHING WRONG WITH MEMORIZATION! IT WORKS! “More intuitive” is good, but I found it perfectly acceptable to be able to get answers BEFORE I completely understood the methods... After I did understand what the “mechanical” method got me, if I decided to go another way, or that I could do something more easily MY way, fine - but why do you want to screw with something that’s not any harder if it has been working for decades?

Millions of people get in a car and drive without the slightest understanding of how the vehicle works. Let’s do something “more intuitive” to teach them how to drive.

It is also worth noting that the quadratic formula did not just mysteriously spring out of nowhere. It was logically derived from the completing the square method.

I taught algebra for quite a few years. In my mainstream classes I just presented the quadratic formula, and mentioned where it came from. Then we applied it. In my honors classes we first took the time to derive the formula.

And here’s where the field of education is different than engineering. Engineers will change things if a new way is better. Educators will change things if a new way is different. No one above the level of teacher bothers to consider if that new way is better or worse.

“Engineers will change things if a new way is better. Educators will change things if a new way is different.”

My Mom taught elementary school for 37 years, and she’s right there with you...

Not sure how to do a virtual fist bump, but consider it done. Have a great Christmas.

Wait a minute! This is useless. Why, because it is wrong to tell the students to get the RIGHT answer because there are no right answers. Just pick two numbers. Or maybe three if you want. Or tell the teacher there is no correct answer.

This guy is racist. He's the one who is making black people beating of Asians. Arrest him.

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