How to robustly identify $\arg \min_{x}(x^TAx+B^Tx+c)$ when $A,B$ and $c$ are varying?

Question

I wish to minimize an expression that looks like this- \begin{equation} e=x^TAx+B^Tx+c \end{equation} where, $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^n$ and $c \in \mathbb{R}$. The problem here is $A$, $B$ and $c$ are varying. What I have done is, I have constructed a quadratic cost, such as, $J=\frac{1}{2}e^2$ and now using gradient descent, \begin{equation} \dot{x}=-\eta\frac{\partial J}{\partial x}=-\eta\frac{\partial J}{\partial e}\frac{\partial e}{\partial x}=-\eta e(Ax+B) \end{equation} The equation that I have derived above is simple GD and it works for a fixed A, B etc and it gives the minimia, i.e., $x_1^*$. But when $A,B$ and $c$ change, this algorithm doesnt yield new $x^*_2$ as it should be. So I want to know, if there is a way to robustly find $x^*$ even when $A,B$ and $c$ change? Any mathematical construct that might help me! Thanks in advance!