Complex Numbers and Quadratic Equations — Formula Sheet

The imaginary unit i is defined as i = √(–1), so i² = –1.

Powers of i repeat with a cycle of 4

i¹ = i, i² = –1, i³ = –i, i⁴ = 1, i⁵ = i, ...

If remainder = 0 → i⁴ᵏ = 1

MULTIPLICATION

(a + ib)(c + id) = (ac – bd) + i(ad + bc) — use distributive law and replace i² = –1.

(z₁ + z₂)² = z₁² + 2z₁z₂ + z₂²

The imaginary unit i is defined as a solution to x² = -1, so i² = -1.

For complex numbers z₁ = a + ib and z₂ = c + id, the sum is z₁ + z₂ = (a + c) + i(b + d).

Subtraction is defined as z₁ - z₂ = z₁ + (-z₂), where -z₂ is the additive inverse of z₂.

For z₁ = a + ib and z₂ = c + id, multiplication is defined as z₁z₂ = (ac - bd) + i(ad + bc).