<s>[INST] Algebra Smallest Possible Value What is the smallest possible value of
$x^2+y^2-x-y-xy$?
Is this even possible to solve? Please help. 
 [/INST] $$2(x^2+y^2-x-y-xy)=(x-y)^2+(x-1)^2+(y-1)^2-1-1\ge-2$$
Alternatively, let $$x^2+y^2-x-y-xy=k\iff x^2-x(1+y)+y^2-y-k=0$$
As $x$ is real, the discriminant must be $\ge0$
$$\implies(1+y)^2\ge4(y^2-y-k)$$
$$\iff4k\ge3y^2-6y-1=3(y-1)^2-3-1\ge-3-1$$ 
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