Complex numbers are numbers that can be expressed in the form of a+bi. Complex numbers are a combination of real and imaginary numbers where a and b are real numbers and i is an imaginary unit. Complex numbers exist because they are needed in order to solve problems seen in geometry, calculus, physics and more.
The Fundamental Theorem of Algebra says all non-constant single-variable polynomials with complex coefficents have at least one complex root. The theorem needs complex numbers in order to make polynomials equal to zero in some cases.
In the link, the axis of symmetry is highlighted in purple. The vertex is marked at the tip of the parabola (0,-1). The origin is marked at (0,0). The discriminant tells the number of real solutions in a quadratic solution. To find the discriminant, you use the expression located under the radical in the quadratic formula. The discriminants value gives information about roots in the polynomial. It is important to the Fundamental Theorem of Algebra because it tells whether the quadratic is irreducible or reducible. If there is a negative discriminant, the quadratic is irreducible.