Why Complex Numbers Exist
Definition
Complex numbers are numbers that can be expressed as "a + bi", where a and b are real numbers and i is an imaginary number.
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra, or FTA, states Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In other words, if there is a second degree polynomial, then there must be exactly 2 roots.
The FTA needs complex numbers in order to be true because the complex numbers are what makes it a theorem in the first place.
Discriminant
A discriminant is the expression inside the radical of a quadratic formula. It relates to the FTA because it determines the possible types of answers. It can either be positive, which means there are two real solutions, zero, so there is one solution, or negative, which gives us two complex solutions.