Gregory XV (Alessandro Ludovisio), pope (1621-1623), born at Bologna in 1554, succeeded Paul V on February 9, 1621. Beyond assisting the German emperor against the Protestants, and the king of Poland against the Turks, he interfered little in European politics. He was a learned divine and manifested a reforming spirit; and his pontificate was marked by the canonization of St Theresa of Avila, Francis Xavier, Ignatius Loyola, and Philip Neri[?]. He died on July 18, 1623, and was succeeded by Urban VIII
... equation, because it discriminates between three qualitatively different cases:
If the discriminant is zero, then there is a repeated solution x, and this solution is ...