Complex Numbers and Quadratic Equations

Imaginary Unit and Powers of i

The imaginary unit i is defined as the square root of -1. Its powers cycle in a pattern of four: i, -1, -i, and 1.

Algebra of Complex Numbers

A complex number is expressed as z = a + ib, where a and b are real numbers. Basic operations like addition, subtraction, multiplication, and division are performed by treating i like an algebraic variable while substituting i squared with -1.

Conjugate and Modulus

The conjugate of a complex number z = a + ib is z-bar = a - ib, and its modulus represents its distance from the origin, given by the square root of (a squared plus b squared).

Argand Plane and Polar Representation

Complex numbers can be geometrically represented on a two-dimensional Cartesian plane called the Argand diagram, and expressed in polar form as r(cos theta + i sin theta).

Solution of Quadratic Equations

When the discriminant b squared minus 4ac of a quadratic equation is negative, the equation yields non-real complex conjugate roots, calculated using the standard formula x = (-b plus or minus square root of D) / 2a.