Put it this way. You start with a quadratic equation, then you manipulate it and turn it into a cubic one. By doing so, youve introduced a stray solution to the equation and it so happens that *x* = −1 is that stray solution. Its like solving an equation involving surds (square roots): if you square both sides to get rid of the square roots, you introduce extra solutions into your equation; hence you need to substitute the solutions youve found back into the original equation to see which ones dont fit.

by dividing both sides by x

And, by substituting to the first equation, you get...

multiplying both sides by x...

finding the cube root...

So now, to check, substitute again...

Which is ,obviously, wrong...

Ok...right?

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The question was, there's a person who who always speaks the truth. There's another who always lies. A stranger comes to the place where there are these two men. The question is, with only one question to ask any of the two, he should be able to tell who's the person who speaks the truth always and who's the liar.

The solution is a bit tricky. The stranger ought to ask a question 'What would the liar say if asked him who's the liar'. If he's the liar, he'd point to himself. That is because, in reality, the liar would have pointed to the truth speaker. But since he doesn't speak the truth, he'd point himself. If he's the one who always speaks the truth, he'd have pointed the liar. Since he does not lie, now he'd point to the liar. Either way, the index finger would be aimed at the liar. This way, the liar can easily be found.

In reality, we come across many contradictory statements. He's a regular late-comer.

There was a law and order failure in the city.

The best I have come across is he is consistently inconsistent. A classic oxymoron!

I know him, he's unpredictable is another.

for complex x:

]]>Got any good contradicting equations? Maybe people could donate such, and others would try to find out what is wrong...

Let's start easy:

(this was in our exam, I thought it was pretty easy for us, until I asked my classmates what they answered...)

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