You teacher is correct in a sense when she said that the square root of a negative number doesn't exist.
You can think about it like this:
When you were just using the numbers 1, 2, 3, 4, and so on back in 2nd and 3rd grade
You'd have something like 1 + ___ = 3, and you'd fill in 2. Or you'd have something like 3 - ___ = 2, and you'd fill in 1.
If someone instead put 3 + ___ = 3, what would you put? None of the numbers 1, 2, 3, and so on satisfy that, so the number that would doesn't exist.
However, later you invented new numbers. You came up with 0 and negative numbers, and suddenly that question had an answer. The answer didn't exist in the old numbers, but it does in the new ones.
Likewise, when you were using just the integers, if someone cut a cupcake into 2 pieces and gave you one, then someone asked you how many cupcakes you were given, you couldn't give them an integer. No number existed that could answer that question. Later though, you defined the rational numbers and you could give a fraction as an answer.
Then you started doing multiplication and division. And at some point, someone asked you what numbers x have the property that x*x = 9, and in the rational numbers, this would be -3 and 3. But what if instead your teacher asked you for the numbers x such that x*x=2? Then you could sit there and try to work it out on paper, and you might be able to find some number that almost worked, but you'd never be able to get an answer, because in the rational numbers, the square root of 2 does not exist, so you invent the real numbers, and they give you numbers that answer some questions like that.
But then someone asks for the numbers x such that x^2 = -9.
Well, you can prove that for any real numbers, if x is not 0, then x^2 > 0, so there can't be any real number x such that x^2 = -9.
Ah, but whenever we had that problem before, we just invented new numbers, so maybe we can just do that again and our question will have an answer. In fact, we can: that is what the complex numbers do.
