[In English, this name derived from Greek may be spelled "Catherine", "Katherine", "Katharine", or "Kathryn". Variants from other languages include "Catalina" (Spanish), "Ekaterina" (Russian), and "Katrina" (German).]
Among the women known to history as "Queen Catherine" are:
Catherine of Alexandria (or Saint Catherine) (~27x-~307), queen regnant of Egypt: daughter of Queen Sabinella of Egypt and King Costis of Egypt, son of Constantius Chlorus
... this case, the parabola does not intersect the x-axis at all.)
Note that when computing roots numerically, the usual form of the quadratic formula is not ideal. See ...