Simple polynomial equations over (2x2)-matrices

We consider the polynomial equationXn + an-1 ⋅ Xn-1 + ⋯ + a1 ⋅ X + a0 ⋅ I = O over (2 x 2)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai ∈ ℝ for each i and n ≥ 2 . We discuss its solution set S supplied with the natural Euclidean topology. We completely describe S. We also show that dim S =2.