Convexity of Quadratic equation Inequality?

Question

Solving an inequality of the form $x^TAx\geq0$ or $x^TAx\leq0$ is straightforward. I mean we have to check if A is positive semidefinite or negative semidefinite. But what would be the solution to the inequality $x^TAx+b^Tx+c\leq0$ and $x^TAx+b^Tx+c\geq0$ ? Specifically I need to know when either of the inequality would be convex. If someone can share a good resource that talks about quadratic equation(not quadratic form) with matrices as coefficients besides Wikipedia, it would be great. Thank you.

Answer 1

If we substitute $x = y + v$ in the inequality $x^T A x + b^T x + c \le 0$ we get $y^T A y + (b^T + 2 v^T A) y + v^T A v + b^T v + c \le 0$. Assuming $A$ is invertible, we can choose $v$ so that $b^T + 2 v^T A = 0$, so the inequality becomes $y^T A y \le k$. If, for example, $A$ is positive definite and $k > 0$, this defines an ellipsoid.