Help with Reduce, quadratic system of inequalities

Question

I'm trying to obtain an equivalent system of inequalities to the one below but without the variables x and z. Somehow Reduce just repeats the command.

v[x_, y_] := a*x^2  + b*x*y + c*x  
sc1 := ForAll[{x, z}, {z >= x && x >= 0 && z <= 1}, v[z, x] - v[x, x] >=0  ]
sc2 := ForAll[{x, z}, {x >= z && z >= 0 && x <= 1}, v[z, x] - v[x, x] <=0  ]
Reduce[(sc1 || sc2), {a, b, c}]

Is there some basic error I can't see? I looked at questions about using Reduce with ForAll but did not find anything similar.

Answer 1

v[x_, y_] := a*x^2 + b*x*y + c*x
sc1 = ForAll[{x, z}, 0 <= x <= z <= 1, v[z, x] - v[x, x] >= 0]
sc2 = ForAll[{x, z}, 1 >= x >= z >= 0, v[z, x] - v[x, x] <= 0]
Resolve[(sc1 || sc2), {a, b, c}]
(*
(a < 0 && ((b <= -2 a && c >= -2 a - b) || (b > -2 a && c >= 0))) ||
(a >= 0 && ((b < -2 a && c >= -2 a - b) || (b >= -2 a && c >= 0)))
*)