The Butterfly Optimization Algorithm (BOA) is a swarm based technique, inspired from mating and food searching process of butterflies, developed in last year. Experiments indicate that BOA provides substantial exploration capability on conventional non-constrained benchmark problems, however for the cases with more complex and noisy domains the algorithm can easily be trapped into local minima due to its restricted exploitation behavior. To tackle this issue, current study deals with introducing an alternative search strategy to explore the region of the search domain with high certainty. Such that, firstly a weighted agent is defined and then a quadratic search is performed in the vicinity of this pre-defined agent. This alternative search strategy is named as Enhanced Quadratic Approximation (EQA) and it is combined with BOA method to improve its exploitation behavior and provide an efficient search algorithm. Thus, obtained new method is named as Enhanced Quadratic Approximation Integrated with Butterfly Optimization (EQB) algorithm. Different properties of proposed EQB are tested on mathematical and structural benchmark problems. Acquired results show that the introduced algorithm, in comparison with its parent method and some other well-stablished reported algorithms in the literature, provides a competitive performance in terms of stability, accuracy and convergence rate.