I have a quadratic minimization problem such that, $e=x^TAx+x^Tb+k$, I wish to find its minima using recursive least square.

Question

I have arrived at a generic quadratic minimization problem, such that, \begin{equation} e=x^TAx+x^Tb+k \end{equation} where,$x=[x_1,x_2,x_3]^T \in \mathbb{R}^3$, $b=[\gamma,0,0]$ and $k$ is a constant. \begin{equation} A=\begin{pmatrix} 0 & 0 & 0\\ 0 & \alpha &0 \\ 0 & 0 & -\beta \end{pmatrix} \end{equation} Let, \begin{equation} x^*=\arg\min_x(e) \end{equation} clearly, $A$ is not full rank. I wish to minimize $e$ over $x$ using recursive least square, is it possible? I know it can be minimized using gradient descent but I want to know if it is feasible to minimize this quadratic using RLS? Also the reason why I am going for RLS is because, gradient descent fails when $A,b$ changes abruptly.