Does the equation x^2=-64 have real solutions?

Of course, not!

There is no
real number that raised to square to give a negative
value.

But the equation has complex
solutions.

x1 = +sqrt(-64) =>x1 =
8i

x2 = -sqrt(-64) => x2 =
-8i

Keep in mind that complex solutions always come in
pairs!

Therefore, the given equation has no
real solutions.