計算機は囁く
Mathematica は当然
In[5]:= Reduce[Exists[x, 0 <= x <= 1 && y == x^2 - 2 a x], Reals] Out[5]= (a <= 0 && 0 <= y <= 1 - 2 a) || (0 < a <= 1/2 && -a^2 <= y <= 1 - 2 a) || (1/2 < a <= 1 && -a^2 <= y <= 0) || (a > 1 && 1 - 2 a <= y <= 0)
と答えます.
一方,qepmax+nnsolvexx は
(%i1) qe([[E,x]],"%and"(0<=x,x<=1,y=x^2-2*a*x)); (%o1) (((a > 0) %and (y+1 > 0) %and (y+2*a-1 <= 0)) %or ((y >= 0) %and (y+2*a-1 <= 0)) %or ((y <= 0) %and (y+2*a-1 >= 0))) %and (y+a^2 >= 0) (%i2) nnsolvexx(%); (%o2) ((a-1 < 0) %and (a > 0) %and (y+2*a-1 <= 0) %and (y+a^2 >= 0)) %or ((y >= 0) %and (y+2*a-1 <= 0)) %or ((y <= 0) %and (y+2*a-1 >= 0))
と囁きます.