When is the 2×2 matrix ring over a commutative local ring strongly clean?

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the 2×2 matrix ring is strongly clean where is the ring of p-adic integers, but is not strongly clean where is the localization of at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for to be strongly clean. As consequences, is strongly clean iff is strongly clean iff is strongly clean iff is strongly clean.