Quadratic equations – Double Solutions, Imaginary Solutions and Just Plain Wrong

Sometimes I find myself lying awake at night, thinking about stuff. Last night, I was kept awake by the realisation that I had forgotten how to solve quadratic equations, which is something which would have been routine for me as a schoolboy.

The quadratic equation is something in this form:

ax² + bx + c = 0  

You get told what a, b and c are, and you have to work out what x is. So I looked it up to remind myself of the formula. It is thus:

x = [-b ± √(b² − 4ac)] / (2a)  

It all came flooding back. The bit inside the square root is known as the discriminant. If it is positive, then there are two possible answers. If it is 0, then there is just one answer. If it is negative, then you get imaginary solutions – i.e. in the weird territory to which the door is the square root of -1.[1]  

SuperGrok gave me some examples, thus:  

What is Wrong about Human Rights

The most depressing thing, it seems to me, about law students today is the number of them who want to go into the area of human rights. It is depressing because the whole concept of human rights, as so many of these students now see it, brings the law into disrepute, is anti-democratic and causes considerable damage to society. Let me explain what I mean.

The expression “human rights” did not appear, much if at all, before 1948, when the Universal Declaration of Human Rights (UDHR) was adopted by the United Nations General Assembly in Paris in 1948. Eleanor Roosevelt was largely to blame, if blame is the right word, but it was accepted by the General Assembly[1]. Some will trace the origin of “human rights” back further than that.  For example the Déclaration des droits de l’homme et du citoyen[2] was adopted by the National Constituent Assembly of France in 1789, during the French Revolution. The cornerstone of that is perhaps to be found in articles 4 and 5:

Article 4

Liberty consists in being able to do anything that does not harm others…

Article 5

The Law has the right to forbid only those actions that are injurious to society. Nothing that is not forbidden by Law may be hindered, and no one may be compelled to do what the Law does not ordain.

These principles are consistent with the notion of minimum interference by the State in the lives of citizens. We will come back to that in a moment. But first, let us consider what is the nature of a human right, in the traditional sense, such as the rights of physical liberty, the right to property and the right of free speech.  These rights

You Don’t Need No Education, says Labour

In Australia, where I live these days, around 1/3 of children go to fee-paying schools. Many Australian families are able to afford these fees, because they are subsidised by the government. In short, the government takes the view that if these schools did not exist, then the government schools would have the burden of educating these children, and so it is only fair to contribute to their costs. There is a pretty broad consensus, I think, that these independent schools do a pretty good job, and generally speaking do a better job than the state schools.

In the UK, in contrast, only around 1/10 of children go to fee-paying schools. School fees are much more expensive, and few people can afford to pay them out of their taxed income. It is not particularly contentious to note that there is, on the whole, yawning gulf in the UK between the standards of the independent schools and of the state schools.

A sensible move in the UK would be to follow Australia’s lead providing government funding for private schools on the basis that the more children who go to private schools, the less the demand on the state schools. A really sensible move would be to couple this with scholarships, so that the most able children from every background have the opportunity for their exceptional talents to be recognised and nurtured. Instead of that, the new Labour government is proposing to impose a 20% tax on independent school fees. There are three inevitable consequences of this disastrous policy:

Sweet ignorance

The children in the UK have been moaning about an exam question, which was as follows:

There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random sweet from the bag. She eats the sweet.

Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3.

Show that n² – n – 90 = 0.

The question isn’t hard. You just take the odds of the first sweet being orange, and then multiply that by the odds of the second sweet being orange, and that is 1 in 3. It is 3 lines of calculation.

But what I think might be quite hard is doing it if you don’t have a piece of paper, but merely a computer. I guess