You can say "there is no solution." But that's boring.

Mathematicians don't like being told they cannot do something. Instead they try to come up with new rules or new definitions to do whatever it is there isn't already a rule for. And those rules can be anything, but generally we look for consistency (those new rules should complement or add on to existing rules), usefulness (the new rules should help us do something we couldn't do before), and interesting consequences (things that make us go "ooh, that's neat").

And the more useful, interesting and neat those rules are, the more likely they are to be used by other mathematicians, and adopted as standard.

With complex numbers, we take all our usual rules for numbers and throw in one more; there exists some number(s) i such that i^2 = -1. It is consistent with all our existing rules, and turns out to be really useful in a bunch of areas of maths and science (and leads to some really interesting results).

i isn't impossible. It doesn't appear on a standard number line, but that's not a huge problem. The number line is an interesting and useful tool, but not the end point of numbers. Interestingly the first mathematical paper to use something like a modern number line was published about the same time Newton was publishing his Principia Mathematica; there weren't number lines when Newton was learning maths. Number lines are fairly modern.

Some classical Greek mathematicians had a very different way of looking at numbers, seeing them more as a lose collection of concepts, with fractions being connections between the different concepts (so 1/2 was a connection between 1 and 2). This did cause them problems, though, when it came to irrational numbers...