Complex Numbers and Quadratic Equations — Formula Sheet

x^2 + 1 = 0

ax^2 + bx + c = 0

z = a + ib

z1 = a + ib, z2 = c + id

z1 + z2 = (a + c) + i(b + d)

z1 - z2 = z1 + (-z2)

z1 z2 = (ac - bd) + i(ad + bc)

i^3 = -i

i^4 = 1

i^5 = i, i^6 = -1

i^-1 = -i, i^-2 = -1, i^-3 = i, i^-4 = 1

i^(4k) = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i

The main objective is to solve ax^2 + bx + c = 0 where D = b^2 - 4ac < 0, which is not possible in real numbers.

To handle such equations, the number system is extended by introducing i, where i^2 = -1.

Two complex numbers z1 = a + ib and z2 = c + id are equal if a = c and b = d.

i^3 = -i and i^4 = 1.

i^5 = i and i^6 = -1.

For negative powers, i^-1 = -i, i^-2 = -1, i^-3 = i, and i^-4 = 1.

It is a solution of x^2 + 1 = 0.

If z1 = a + ib and z2 = c + id, then z1z2 = (ac - bd) + i(ad + bc).