Complex Numbers and Quadratic Equations

Imaginary Unit and Complex Numbers

The square root of a negative number introduces the imaginary unit i, where i squared equals negative one. A complex number is expressed in the form a + bi, where a is the real part and b is the imaginary part.

Algebra of Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided similarly to binomials, keeping in mind that i squared is replaced by negative one.

Modulus and Conjugate

The modulus of a complex number z = a + bi is the distance from the origin, given by the square root of (a squared plus b squared). Its conjugate is obtained by changing the sign of the imaginary part to get a minus bi.

Argand Plane and Polar Representation

A complex number can be geometrically represented as a point on a plane called the Argand plane. It can also be expressed in polar form using its modulus and argument (angle).

Quadratic Equations with Negative Discriminant

When the discriminant b squared minus 4ac of a quadratic equation is negative, the quadratic formula yields complex roots involving imaginary numbers.