We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to $15.51560$, and the efficiency index is raised from $1.6817$ to $1.9847$. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.