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1. z^{2} = -1

   Start from the equation as given.

2. z^{2} + 1 = 0

   Bring everything to one side and collect like terms so the right-hand side is 0.

3. a = 1,\quad b = 0,\quad c = 1

   Read off the coefficients of the standard form ax² + bx + c = 0.

4. \Delta = b^2 - 4ac = (0)^2 - 4(1)(1) = -4

   It does not factor nicely, so use the discriminant Δ = b² − 4ac.

5. z = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{0 \pm \sqrt{-4}}{2}

   Δ < 0, so the roots are a complex-conjugate pair. Substitute into the quadratic formula.

6. z = i \quad\text{or}\quad z = - i

   Simplify each root.