Positivity of a quadratic expression

Completing the square.
$$x^2+xy+y^2=\left(x+\frac y2\right)^2+\frac34y^2.$$

Non-negativity. Both terms are $\ge0$. The first is $\ge0$; the second is $\ge0$.

Vanishing. The expression equals $0$ iff $x+y/2=0$ and $y=0$, i.e. $(x,y)=(0,0)$.

Conclusion. The quadratic form is positive definite.