1. The problem states: Solve the equation $x^2 = -16$. 2. Recall that the square of a real number $x$ is always non-negative, so $x^2 = -16$ has no real solutions. 3. However, in the complex number system, we can solve this by introducing the imaginary unit $i$ where $i^2 = -1$. 4. Rewrite the equation as $x^2 = -16 = 16 \times (-1) = 16i^2$. 5. Taking the square root of both sides, we get $$x = \pm \sqrt{16i^2} = \pm \sqrt{16} \sqrt{i^2} = \pm 4i.$$ 6. Therefore, the solutions are $x = 4i$ and $x = -4i$.