For which values of $k$ the Inner product is well defined

Question

set Inner product with $\mathbb{R}^2x\mathbb{R}^2\longrightarrow\mathbb{R}$
$<(x_1,x_2),(y_1,y_2)>=x_1y_1-3x_1y_2-3x_2y_1+kx_2y_2$
For which values of $k$ the Inner product is well defined?>

Hi there , I have this question and I'm not sure I solved it right.
When checking for "Positive semi-definite" I get -

$<(x_1,x_2),(x_1,x_2)>=x_1^2-6x_1x_2+kx_2^2 \ge 0$

Here I'm not sure what to do. If I do Trinomial I get that for $k\ge9$ the equation is $\ge0$ for all $x_1,x_2\in \mathbb{R}$.
So this is the solution? I'm not sure because if $0<k<9$ then for some $x_1,x_2\in \mathbb{R}$ in might be true and for some it might be false...

Thank you in advance!