Prove that $x^2 + 5xy+7y^2 \ge 0$ for all $x,y \in\mathbb{R}$ This is probably really easy for all of you, but my question is how do I prove that $x^2 + 5xy+7y^2 \ge 0$ for all $x,y\in\mathbb{R}$
Thanks for the help!

Another possibility is to note that the quadratic form is represented by the matrix
$$ \begin{pmatrix}  1 & 2.5 \\ 2.5 & 7\end{pmatrix} $$
The claim follows from the fact that its eigenvalues are both positive, so that the matrix is positive definite.
