Solving non-commutative "quadratic" equation with inhomogenously typed coefficients

Question

Is there a general method to solve for $z\in\mathbb{R}^d$ in the non-commutative $z^\intercal\alpha z + \beta z + \gamma = 0$ where $\alpha\in\mathcal{M}_d(\mathbb{R})$ (real $d\times d$-matrix), $\beta \in \mathbb{R}^d$ (real $d$-vector), and $\gamma\in\mathbb{R}$? It looks partially quadratic, but is non-commutative, and has coefficients of different types.