The Quadratic Formula

Theorem: The roots of a quadratic polynomial  P(x) = a x 2 + b x + c  are located at

The Fundamental Theorem of Algebra tells us that a quadratic polnomial may have  2, 1, or  0  roots. The case depends entirely on the discriminant: the term  b ² – 4 ac  in the quadratic formula.

b 2 – 4 ac > 0 . There are two roots, +  or  –  the positve square root.
b 2 – 4 ac = 0 . There is one root, +  or  –  0  leads to the same root. (The root has multiplicity 2. See The Factor Theorem.)
b 2 – 4 ac < 0 . There are no roots, the square root is undefined. (The roots are complex numbers.)