3.D.4.a. Cubic Equation

The cubic equation

        x 3 +a x 2 +bx+c=0

has 3 roots given by

        x 1 =S+T 1 3 a

        x 2 = 1 2 ( S+T ) 1 3 a+i 3 2 ( ST )

        x 3 = 1 2 ( S+T ) 1 3 ai 3 2 ( ST )

where

        S= R+ Q 3 + R 2 3          T= R Q 3 + R 2 3

        Q= b 3 ( a 3 ) 2                    R= a 3 b 2 c 2 ( a 3 ) 3

The character of the roots is determined by the discriminant

        D= Q 3 + R 2

so that

        D>0          Þ                  1 real & 2 complete conjugates

        D=0          Þ                  all real with at least 2 equal

        D<0          Þ                  all real & all equal