1. **State the problem:** Solve the equation $x^2 = -16$. 2. **Recall the formula and rules:** To solve $x^2 = a$, we take the square root of both sides: $x = \pm \sqrt{a}$. 3. **Apply the formula:** Here, $a = -16$, which is negative. The square root of a negative number involves imaginary numbers: $\sqrt{-16} = \sqrt{16 \times (-1)} = \sqrt{16} \times \sqrt{-1}$. 4. **Simplify:** $\sqrt{16} = 4$ and $\sqrt{-1} = i$ (the imaginary unit). 5. **Write the solution:** $$x = \pm 4i$$ **Final answer:** $x = 4i$ or $x = -4i$.