eliminate

i1 : R = ZZ/101[x,a,b,c,d]   

o1 = R

o1 : PolynomialRing

i2 : f = x^2+a*x+b

      2
o2 = x  + x*a + b

o2 : R

i3 : g = x^2+c*x+d

      2
o3 = x  + x*c + d

o3 : R

i4 : time eliminate(x,ideal(f,g))
     -- used 0.004 seconds

                      2    2             2           2
o4 = ideal(a*b*c - b*c  - a d + a*c*d - b  + 2b*d - d )

o4 : Ideal of R

i5 : time ideal resultant(f,g,x)
     -- used 0.004 seconds

                        2    2             2           2
o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )

o5 : Ideal of R

i6 : sylvesterMatrix(f,g,x)

o6 = {0} | b 0 d 0 |
     {1} | a b c d |
     {2} | 1 a 1 c |
     {3} | 0 1 0 1 |

             4       4
o6 : Matrix R  <--- R

i7 : discriminant(f,x)

        2
o7 = - a  + 4b

o7 : R