Quadratic Equations

Now they tell us – a new and simpler way to solve quadratic equations.

Photo by Lewis Mulatero/Getty Images

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 fo

r two identical values to begin with. This simplifies the arithmetic part of multiplying the formula out.

Photo by Popular Mechanics

“Normally, when we do a factoring problem, we are trying to find two numbers that multiply to 12 and add to 8,” Dr. Loh said. Those two numbers are the solution to the quadratic, but it takes students a lot of time to solve for them, as they’re often using a guess-and-check approach.

Instead of starting by factoring the product, 12, Loh starts with the sum, 8.

If the two numbers we’re looking for, added together, equal 8, then they must be equidistant from their average. So the numbers can be represented as 4–u and 4+u.

When you multiply, the middle terms cancel out and you come up with the equation 16–u2 = 12. When solving for u, you’ll see that positive and negative 2 each work, and when you substitute those integers back into the equations 4–u and 4+u, you get two solutions, 2 and 6, which solve the original polynomial equation.

It’s quicker than the classic foiling method used in the quadratic formula—and there’s no guessing required. —Courtney Linder

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.

“Net Zero” Emissions – Baloney | The Economist

A gateway to endless fudge – that’s what The Economist calls various countries’ commitments to “net zero” emissions (“The power of negative thinking” in the May 8 issue).
I was delighted to read that because for the longest time I’ve thought that the “net zero” concept was baloney but The Economist puts it much better and explains why. “It allows the ultimate scope of emission cuts to remain undefined and sweeps all the uncertainties under a carpet of techno-optimism.

Source: How to make long-term climate pledges add up | The Economist

“IBM built a system that can’t read barcodes”

Dr. Robert Van Exan, former director of health and science policy at Canadian pharmaceutical giant Sanofi Pasteur, said tracking with barcodes in Canada “should have been written in the pandemic plan.”There was a plan to make these barcodes central to Canada’s public-health system, and there was a time when Canada was ahead in digitizing its health system “by a decade,” Dr. Van Exan said. Canada’s 1998 vaccine strategy first proposed barcoding vaccines to promote efficiency and accuracy. The 2003 SARS epidemic, and the creation of the Public Health Agency of Canada, hastened that work.In normal times, Canada administers millions of vaccines a year for diseases such as mumps and influenza. Provinces slowly adopted digitized immunization records in the early 2000s, but continued entering all the data manually: Audits of some provincial systems found fully 15 per cent of immunization records were incomplete, nearly a quarter had inaccurate information, and crucial data was missing from one in five adverse-reaction reports.In 2007, Ottawa tapped an advisory group made up of industry experts, including Dr. Van Exan, to plan the implementation of these barcodes. The total cost, the advisory group found, would have then been around $265-million, but they projected savings of $1-billion in the decades to come. They handed Ottawa a plan to start barcoding vaccines in warehouses, hospitals, clinics and pharmacies by 2014.This barcoding capability was an integral part of a broader digital infrastructure project known as the Vaccine Identification Database System (VIDS). Ottawa set up VIDS as a proof of concept for a single, national digitized public-health system to track infectious disease outbreaks and vaccination campaigns.Story continues below advertisementOttawa contracted IBM Canada to build a permanent vaccination version of VIDS, called Panorama. That’s where things “fell off the wagon,” Dr. Van Exan said. “IBM built a system that can’t read barcodes.”Beset by delays and cost increases, some provinces dropped the project. Even some provinces that stuck with Panorama have still not installed crucial components of the system. None of the provinces’ systems work with one another.“This is one of the big flaws in the whole damn system,” Dr. Van Exan said.

Source: Canada falls behind on barcode technology for COVID-19 vaccine tracking – The Globe and Mail

Mitch McConnell on Leadership

“Mr. McConnell had considered voting to convict the former president as a means of purging him from the party, but allies said he concluded he could not practically, as leader, side with a minority of his colleagues rather than the overwhelming number who said the trial was invalid and voted to acquit. Instead, he used every ounce of his rhetorical strength to try to damage Mr. Trump’s credibility with his own party. “
As if leadership consists of knowing which way the wind is blowing.  McConnell, as much as Trump, is responsible for the breakdown in good governance in Washington. Maybe more so, because if McConnell had had the guts and the decency to call, “Liar”, four years ago a lot of this could have been avoided.
I’m so mad I could spit.

I’m getting a bit ticked at all these Covid patients…

A few people  dying and a lot more getting sick are spoiling things for the rest of us.
I think the various States of Emergency should do something positive and effectual instead of fining people for attending parties and not wearing masks.
Here’s my idea: enact legislation that gives the Government of Ontario the right to access and integrate data from patients’ smartphones with Covid patient data from OHIP and add in other location data from Presto, the Beer Store, LCBO, Ontario Lottery and Gaming, universities and school boards. Throw all the data in a big tub and turn on the Artificial Intelligence.  The output would be a demographic description of the people who are spoiling things.  Lock them down, tightly. Problem solved, more or less.


I think the trade-offs between privacy and health/safety are due for a re-balancing. I can’t help thinking that there’s a logical answer to the question, “Who are all these people?”.  An analogy: investigators of fatal traffic accidents will often report whether the injured parties were wearing seatbelts or whether excessive speed was involved or whether alcohol was a factor.  We should have Covid-19 investigators who report what contributing factors the patients have.  A possible downside: some symptomatic people would delay getting tested if a positive result would “out” them….a trade-off worth making, I think.

Piketty and the decline of “dirty mansions” – Spacing Toronto

This is very interesting – an explanation of the economics behind the increase in big, big houses in some Toronto neighborhoods, the subsequent conversion of many of them to apartments and rooming houses, and the more recent restorations back to single-family homes.  The author, Dylan Reid, attributes it to cycles of income and wealth disparity: the decrease in single family occupancy of the big homes after WW 2 was due to reduced inequalities – the owners couldn’t afford to maintain their wealthy lifestyles.  He backs up his argument with economic data from that era.
Recent gentrification is a result, Reid suggests, of increasing disparities of wealth and income.
I wonder what part of the changes in income inequality is due to the Baby Boom.  For that I’d have to read Piketty’s original work and it’s a bit over my head.


Toronto’s affordable housing crisis has many facets. One of these many facets is the conversion – or rather, re-conversion – of big old houses in the older parts of the city that are full of relatively affordable rental apartments or rooms, back into single-family, owner-occupied homes. It’s a process that is no doubt affected by […]

Source: REID: Piketty and the decline of “dirty mansions” – Spacing Toronto

Trump cancels Republican convention in Florida after coronavirus spike – Reuters

Screwy and typical of Trump, but where’s the Republican National Convention in all this? I’m wondering if Trump overstepped his bounds by pulling the plug on the Jacksonville convention. It’s the RNC’s convention, not his. If he announced the cancellation without the blessing of the RNC he will have burned up a lot of good will with the GOP establishment.

Source: Trump cancels Republican convention in Florida after coronavirus spike – Reuters

Anxiety? Depression? or just paying attention?

A third of Americans now show signs of clinical anxiety or depression, Census Bureau finds amid coronavirus pandemic



The U.S. Census Bureau has been tracking ​adult ​American’s responses to the pandemic including some questions relating to mental health.​ https://www.cdc.gov/nchs/covid19/pulse/mental-health.htm (with neat drop-down filters). ​

The results purport to measure some indicators of mental health​ wellness​. I think the results are more an indicator of who’s paying attention The charts are based on respondent answers to these questions:

Over the last 7 days, how often have you been bothered by … having little interest or pleasure in doing things? Would you say not at all, several days, more than half the days, or nearly every day?

Over the last 7 days, how often have you been bothered by … feeling down, depressed, or hopeless? Would you say not at all, several days, more than half the days, or nearly every day?

Over the last 7 days, how often have you been bothered by the following problems … Feeling nervous, anxious, or on edge? Would you say not at all, several days, more than half the days, or nearly every day?

Over the last 7 days, how often have you been bothered by the following problems … Not being able to stop or control worrying? Would you say not at all, several days, more than half the days, or nearly every day?

“Feeling down”? “Feeling nervous”? I’d say that anyone who doesn’t acknowledge feeling down and feeling nervous hasn’t grasped the seriousness of the situation. I’m guessing that the two-thirds of Americans who are not showing signs of clinical anxiety or depression aren’t wearing face masks.

The breakdowns of the responses aren’t what I would have guessed. I thought people with more years of formal education would report more feelings of anxiety or depression. Wrong – it’s respondents with less formal education. And I thought younger respondents would report feeling more anxious/depressed than older people. Just the opposite.


“Totally unacceptable…”. Who writes this stuff?

With school boards across the continent struggling with planning for the most tumultuous school reorganization in anyone’s memory, it seems a bit far-fetched to protest possible reductions to French immersion


L’Assemblée de la francophonie de l’Ontario denounced talks about French programs in a tweet Wednesday night, calling it “totally unacceptable.”

“In a bilingual country, to propose the elimination of the teaching of one of the official languages is unthinkable,” the organization representing Franco-Ontarians wrote. “We also ask the Toronto District School Board to stop using official languages as a means of pressure to reach its goals. It’s damaging for the country.”

Coronavirus and the crisis of capitalism

There’s something between the lines here…I agree with the author’s premise that fiscal policy to deal with the pandemic could lead to a Universal Basic Income but he muddies the waters when he turns philosophical and wonders about the future of capitalism.  I’d like to know how creative programs to deal with the pandemic could reform conservative economic thinking.  Don’t call it Universal Basic Income. Call it …?  That’s what I want to know… how to present and package an effective safety net in a package that Republicans could support.

Source: Coronavirus and the crisis of capitalism