Square-root splitting of a double polynomial root ($x-x_0=O({ϵ}^{1/2})$)

Suppose a perturbed polynomial has the local form $A(x-x_0{)}^2+ϵB+O((x-x_0{)}^3,ϵ(x-x_0),{ϵ}^2)$ with $AB\ne 0$. Dominant balance gives $x-x_0={ϵ}^{1/2}z$ and $A{z}^2+B=0$. The two resulting Puiseux series split a double root; their leading displacements can be real or a complex conjugate pair. Ordinary Taylor series in $ϵ$ cannot describe this splitting.