Using real numbers, the equation #x^2=-1# has no solutions. In fact, squaring a number means to multiply it by itself. And if you multiply a positive number by itself, you're multiplying two positive numbers, and the result is positive. If you're squaring a negative number, you have the multiplication of two negative numbers, which is positive.
So, either way, a square cannot be negative.
In order to solve this equation, mathematicians have invented a new set of numbers, called complex numbers. The main innovation is the introduction of the imaginary unit, #i#, with the property that
So, in a certain sense, #i# is the square root of #-1#. In this new number set, all quadratic equations have solutions (actually, all polynomial equations do). In fact, the equations which were impossible to solve with real numbers (like the one we started with) can now be solved as
#x^2 = -1 \iff x = \pm sqrt(-1) = \pm i#