1. **State the problem:** Solve the equation $x^2 = 1$ for $x$. 2. **Formula and rules:** The equation is a quadratic equation in the form $x^2 = a$. To solve for $x$, take the square root of both sides, remembering that both positive and negative roots are possible. 3. **Apply the square root:** $$x = \pm \sqrt{1}$$ 4. **Simplify the square root:** $$x = \pm 1$$ 5. **Final answer:** The solutions to the equation $x^2 = 1$ are: $$x = 1 \text{ or } x = -1$$