x^2 = -1
(You probably know x^2 means x squared, but I just thought I'd check!)
At this point you are probably thinking, "but you can't have a negative square number," which is absolutely right, if you are dealing with real numbers. So the answer to the question of how to solve the equation is to not look at real numbers. This is where imaginary numbers come in.
An imaginary number is represented by the letter i. i is equal to the square root of -1, so as you can probably guess, the answer to the above equation is in fact:
x = i
This is a very strange concept for some people, because imaginary numbers are not something that you can imagine in real life. Whole numbers, fractions and even irrational numbers are easily translated into real life problems, but imaginary numbers are an abstract concept.
There is another type of number that uses imaginary numbers. These are called complex numbers. These are simply numbers of the form a + bi, where a and b are real numbers. All previous numbers I have talked about are complex numbers too. If we take a or b to be equal to zero (in maths, or usually means one or both) then we can see how real numbers and imaginary numbers are both of the same form as complex numbers.
Complex numbers can create some very beautiful things. For example, take the Mandelbrot Set.