Thursday, December 12, 2019

This Is Not A New Way To Solve Quadratic Equations

There are no "tricks" in math.  Sure, there are plenty of things that are not intuitive (probability calculations often fall into that category), and some proofs may require an insight that is not obvious, but there are no "tricks".  "Trick" implies some cheat or shortcut; in math, proving theorems requires every step to be mathematically valid--thus, no "tricks".

So when I started to read this article about "a new way to make quadratic equations easy", I was at first taken aback by the fact that the author's repeated claim that completing the square, a crucial step in a common derivation of the quadratic formula, was a "trick".  Still, I read on.

The author explained a supposedly new way to solve quadratic equations.  I found it no more intuitive than the "repeating the square" derivation and use of the quadratic formula, and it relied on an insight that would not be obvious to an Algebra 1 student:
Multiplying out the right hand side gives


This is true when -B=R+S and when C=RS.

Now here comes the clever bit. Loh points out that the numbers, R and S, add up to -B when their average is -B/2.

“So we seek two numbers of the form -B/2±z, where z is a single unknown quantity,” he says.

Why must R and S add up to -B when their average is -B/2?  I of course know why, but an Algebra 1 student isn't naturally going to see it.

Here's the example given:

Is that truly any easier than the standard quadratic formula?  And is this "new" method truly any easier, especially when a is not equal to 1?
Attempting the same problem using the traditional method is much trickier. Go on, give it a go! The new approach is much easier and more intuitive, not least because it doesn’t require the formula to be memorized at all.
Does it not require memorization?  As quoted in the picture above, does it not require memorization of the fact that "The first step is to think that the two roots of the equation must be equal to -B/2±z = 1±z"?  Where is the simplicity?

Then came this:
Loh says he "would actually be very surprised if this approach has entirely eluded human discovery until the present day, given the 4,000 years of history on this topic, and the billions of people who have encountered the formula and its proof. Yet this technique is certainly not widely taught or known."
Complete and total crap. The axis of symmetry of a parabola (in the given form) is x=-b/2a. The roots must be equidistant from the axis of symmetry (because the curve is symmetric about this line, duh); this is given in the quadratic equation by the -b/2a +/- root(b-squared - 4ac)/2a. To clarify, that's -b/2a plus some amount, and -b/2a minus that same amount.  In other words, there's nothing at all "new" that's been "discovered" here--just a slightly different way to think about what every math major (and most Algebra 2 students) already knows.  This method is certainly no easier for students to comprehend--again, the example above looks easy only because a=1.  Not all quadratic equations have such a simple leading coefficient!
“Perhaps the reason is because it is actually mathematically nontrivial to make the reverse implication: that always has two roots, and that those roots have sum −B and product C,” he says.
Again, well-known.  In fact, give me any two roots of a quadratic, and I can tell you the equation of the simplest quadratic with those roots because the sum of the roots is -b/a (again, Loh oversimplifies by assuming a is equal to 1) and the product of the roots is c/a.  I've taught that for years.

Example: a quadratic has roots 2+/-5i.  The sum is 4, the product is 29.  If a=1, then b=-4 and c=29.  Thus, the simplest quadratic that has those roots is x^2-4x+29.  Not rocket science.

Loh has discovered nothing new here.  His method isn't "widely taught" because it's not as efficient (or indeed as elegant) as the standard approach:  derive the quadratic equation by completing the square, then use the quadratic formula to solve quadratic equations.  This "discovery" smacks of someone trying to make a splash and get some acclaim, sort of like those who claim that we should not use pi but instead should use tau (which equals 2*pi).  Silly, unnecessary, unhelpful.


Unknown said...

I'm not hugely mathy, but when I saw the article, right away thought it was just a variation on r+s=-b/a, etc. I sent it to my math PhD brother who agreed it was nothing new.

Totally overhyped and silly.

-- Ann in LA

Anonymous said...

Isn't math racist or something?

Here's the proposed framework for this woke math class: