Perturbation Method

x² + εx - 1 = 0, where ε << 1
Unperturbed: (ε = 0): x² - 1 = 0 => x₀ = ± 1
Perturbed: Assume x = x₀ + ε x₁
Substituting into x² + εx - 1 = 0 yields
(± 1 + εx₁)² + ε(± 1 + εx₁) - 1 = 0
Neglecting higher order terms, roots are
x = 1 - ^ε⁄₂ + ^ε2⁄₈
and
x = -1 - ^ε⁄₂ - ^ε2⁄₈