1. **State the problem:** Solve the equation $z^2 - 1 = 0$ and represent the solutions on an Argand diagram. 2. **Formula and rules:** The equation is a quadratic in complex number $z$. We use the difference of squares factorization: $$z^2 - 1 = (z - 1)(z + 1) = 0.$$ The solutions are the roots where each factor equals zero. 3. **Solve for $z$:** - Set $z - 1 = 0 \Rightarrow z = 1$ - Set $z + 1 = 0 \Rightarrow z = -1$ 4. **Interpretation:** The solutions $z = 1$ and $z = -1$ are points on the complex plane (Argand diagram) located at $(1,0)$ and $(-1,0)$ respectively. 5. **Argand diagram representation:** Plot the points $1 + 0i$ and $-1 + 0i$ on the real axis. **Final answer:** The solutions to $z^2 - 1 = 0$ are $$z = 1 \text{ and } z = -1.$$