I'm a computer science graduate and I'm not sure to which level of Algebra this question belongs to, whether it is abstract algebra or some other filed, but I'm curious to know and learn.
First, before the usage of negative numbers in multiplication, multiplication was defined as repeated addition of positive numbers(or integers), so we can have the equation $$x^2 -4x + 3 = 0$$
Where I mean by the multiplication in $x^2$ and $4x$ multiplication as repeated addition, I know nothing about negative numbers multiplication yet.
We can rearrange the terms and factorize so we have:
$$(x-2)(x-2) = 1$$
Now, if we think of multiplication as repeated addition, with no mention of negative numbers multiplication, then $(x-2)$ can only be positive, so we have $(x-2) ^2 = 1$ so we have one solution, namely, $x = 3$.
But for me, here comes the problem: there is another solution, namley $x = 1$, and we can find this solution only in case we allowed $(x-2)$ to be negative number, and defined multiplication of negative numbers by negative numbers so for example
$$-1 * -1 = 1$$
in which case $(x-2) = - 1$ and we have $x = 1$.
So how come that we first dealt with multiplication as repeated addition, but then we needed to define(or extend) it in another way so that we can get all the solution of an equation that contained terms multiplied in the sense of repeated addition only?
Can someone recommend a textbook that explain this and related stuff from the beginning, illustrating the motivation and building on this possibly also for complex numbers as well? Again I'm not sure in which branch or level of Algebra these stuff are explained.
By the way my question isn't: "why is multiplication of negative numbers defined as it is", which answer is: "to preserve the structure", but rather why must we consider multiplication of negative numbers in the mentioned example equation to arrive at all solutions ! That is it seems that multiplication using negative numbers in the mentioned equation is so inherent and fundamental. I'm not sure if I phrased my question correctly but here it goes...