There is a bug in functionality of “Reduce” in Mathematica 9:
In[10]:= Reduce[3 x^2 + 8 x y + 2 y^2 == 0, {x, y}, Integers]
Out[10]= False
Which is evident that there is a solution, namely:
x == 0 && y == 0
On the other hand:
In[9]:= Reduce[3 x^2 + 8 x y + 2 y^2 == 0, {x, y}, Rationals]
Out[9]= (x | y) \[Element]
Rationals && ((x <
0 && (y == -2 x - Sqrt[5/2] Abs[x] ||
y == -2 x + Sqrt[5/2] Abs[x])) || (x == 0 &&
y == 0) || (x >
0 && (y == -2 x - Sqrt[5/2] Abs[x] ||
y == -2 x + Sqrt[5/2] Abs[x])))
It is not able to solve such a trivial equation, and it thinks like there may be many solutions, while again x=0, y=0 is the only solution also for Rationals.