The equation \( x^2 = -9 \) involves finding the square root of a negative number. In the realm of real numbers, there are no solutions for \( x \) because the square of any real number is always non-negative.
However, in the field of complex numbers, the equation can be solved. The solutions would be \( x = 3i \) and \( x = -3i \), where \( i \) is the imaginary unit.
Thus, the equation \( x^2 = -9 \) has 2 solutions in the complex number system, and 0 solutions in the real number system. If the question assumes a context of real numbers, then the answer would be 0 solutions. If considering complex numbers, it would be 2 solutions.