Show that $V=\{x\in R^n : \langle Ax,x\rangle+\langle b,x\rangle+c=0\}$ is an $(n-1)$-dimensional $C^\infty$ manifold, if $A$ is symetric and inversible and $\theta=\langle b,A^{-1}b\rangle-4c \in R-\{0\}$.
I've tried to show that $g(x)=\langle Ax,x\rangle+\langle b,x\rangle+c$ is a submersion, i.e. $g^´(x)h=\langle 2Ax+b,h\rangle$ is a surjective linear transformation for all $x$. But i didn't find something good and i do not know how to use the hypotesis of $\theta$.
Sorry for my naivety, i'm just a biginner in analysis. Any hint?
The first step is algebra: try to "complete" the square and write your function as $$\langle A(x+ x_0), x+x_0\rangle - d$$ for some $x_0$ and $d$ that are to be determined. Now show that the condition given is equivalent to $d \ne 0$.
Now reduce to the problem: $$Q(x) = \langle Ax , x\rangle = d$$ gives a submanifold if $d\ne 0$. At this step the only thing you use is that $Q$ is homogenous ( of degree $2$, but it works for any degree).