Happy New Persian Year Nowruz if you are celebrating on March 21!

In our previous post on common mistakes with square roots we have dethroned the popular myth that \(\sqrt{x^2}=x\). Below we shall tell our readers a personal story how we gained a life-changing experience due to the basic fact that \(\sqrt{x^2}=|x|\). Students have told us that the absolute value \(|x|\) is taught in the UK only at A levels (age 17-18). Moreover, the last year STEP examiners’ report again highlighted this common mistake about \(\sqrt{x^2}\). So we have shown the graph \(y=|x|\) in the picture above just in case.

### Entrance exams in Physical-Mathematical Schools

We started to learn maths at overseas schools that specialise in mathematics and physics. All such schools are state-funded, but have rigorous entrance exams, usually at the age of 12 or 13. Our traditional riddle below contains Question 2 (of 8) from a recent entrance exam.

So we were in a class of 30 best maths students selected in a 1-million city. Everyone was a maths champion in their previous school. However our new Physical-Mathematical School was rather different. Grades were not given for past successes or for social connections, but only for hard independent work.

### Answering in front of a class

Solving unexpected problems at a blackboard in front of a class was a routine exercise every lesson unless we had a written paper for independent solving. Once the teacher called one of the prettiest girls, who might have been named as a class queen in the UK. At that time she actually was in the top 20% of the class due to her background knowledge from previous years.

While the girl was trying to solve a new problem at the board, our teacher realised that the girl confuses some basic things. So she directly asked: “What is the square root of \(x^2\)?” The girl replied: “x”. The teacher: “That’s a fail, sit down.”

There was complete silence, nobody laughed. Indeed, despite we were the best in our previous schools, I’m sure that everyone from our class had been in a similar sutation before, probably with more advanced stuff than \(\sqrt{x^2}\), however we understood all feelings anyway. The key lesson in this story is not about \(\sqrt{x^2}\) at all, but about the exemplary reaction to such a failure.

The girl hold the nerves and quietly sat down and was actually ok after the lesson. The whole class was really impressed by her level of self-control at the age of 13. I’m not sure if it was a life-long lesson for the girl, but definitely for me since I can vividly recall all the details more than 20 years after this lesson. Later when I was in many tough situations I remembered how to properly *withstand a punch with dignity and never give up!*

### There is no success without failures

Without our failures at school, we wouldn’t become professional mathematicians, but this is another story for our future post. The key arguments are outlined in the article The science of success by NewScientist. You may also enjoy the recent success story of the new billionaires behind *WhatsApp* from BBC. Here is a couple of the very inspiring quotes: “Got denied by Twitter HQ. That’s ok. Would have been a long commute.” “Facebook turned me down. It was a great opportunity to connect with some fantastic people. Looking forward to life’s next adventure.”

It is impossible to imagine in the UK anything similar to the school story above. Students are usually not called to solve unexpected problems in front of a class, even at university. Many vice-chancellors are proud of a 2% dropout rate at their universities. At the opposite extreme, we can mention Bear Grylls, who was one of 4 (among 180) candidates selected for SAS.

### Weeping because of Pythagoras’ theorem

When we were running summer schools for the gifted and talented in the UK, we knew that students should not be asked, at least if they prefer to hide. Once we were discussing Pythagoras’ theorem with year 9-11 students (of age 14-16). Most students knew at least the first half of full Pythagoras’ theorem even if without a proof. However, one girl who was hiding quite well unfortunately decided that Pythagoras’ theorem is beyond her ability and started to cry.

This accident has taught us that we should teach Pythagoras’ theorem inĀ a much more careful way. That is why we have designed our free tutorial on Pythagoras’ theorem including its full statement, proofs in both directions and applications to diaphontine equations. Actually, our riddle on a proof of converse Pythagoras’ theorem is still open for more than 6 months. Hence those students who have enough motivation to self-register and master our free tutorial, can still solve this Pythagoras’ riddle and gain access to other tutorials on advanced topics.

### Your personal choice: crying or learning

You could make your own conclusions from the cultural differences described above. We shall highlight only our own personal choice. We are human and also make mistakes, which is the human nature. Even robots fail and actually more frequently than humans. However, every failure makes us only stronger, because we learn not to repeat the same mistake again. And the harder we fail, the stronger we become. So we call all our failures the important life-changing experience. Here is the powerful principle: *the harder the training, the easier the exam!*

Our riddle is from an entrance exam to a Physical-Mathematical School at the age of 12-13.

**Riddle 13**: factorise the polynomial \(6m^4-3m^3-4m^2+1\).**How to submit**: to write your full answer, submit a comment.**Hint**: any factor of a polynomial gives a root and vice versa.**Warning**: justify that you found all factors with real coefficients.**Prize**: free 1-year access to one of our interactive web tutorials.**Restriction**: only the first correct public answer will be rewarded.**Update**: Chen has excellently solved the riddle at first attempt.

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