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### Lesson 1

#### The subject of the lesson

In this lesson we're going to solve some second degree equations in one unknown, showing that some equations have solutions, while others don't. Then we'll solve some linear system in two unknowns. All these concepts are very important in the path to the introduction of the set of complex numbers.

#### The lesson begins

If we consider the equation x2-1=0, we can immediatly find the solutions, that are x=±1. A slight modification of  the equation, x2+1=0, provides an example of a problem with no real solution.

This is somewhat cumbersome: two very similar problems have a completely different behaviour. Mathematicians don't like this and they have found the way to overcome the problem. We'll see in the next lesson how this is possible.

Now try to find some other examples of the same kind: an equation has solutions, while an almost similar one has no solution.

Now we are interested in the problem of solving a system of two equations in two unknowns. If the system is a first degree one, its solution is straightforward (using for instance a substitution technique), but if it is a second or higher level degree, the solution is not always simple. We'll now consider some examples and try to solve them, in order to bring to our minds the main techniques.