<p>This paper aims to address an intrinsic relationship between some well-known contractions by proposing a novel metrizability method that converts some well-known contractions into a Banach contraction, preserving the completeness of the underlying metric space. Adding a new metric that does not change the original space shows that the Hardy-Rogers contractions can be observed as Banach contractions. This gives us a new way to look at these fixed point theorems. We substantiate the theoretical findings with appropriate examples that highlight the autonomy of contractive conditions, emphasizing the novelty of the proposed methodology. In addition, the convergence rate and factors of iterative methods are analyzed within the new metric space, accompanied by a numerical example. that illustrate the practical advantages of the methodology. We provide an application to identify the existence of a solution for the Lane-Emden Equation, which is relevant to galactic structure and polytropic star simulations, to support the presented results.</p>