This empirical study investigates the impact of two distinct mathematical problem-solving methods – the Algebraic Formula Method and the Newton Sum Method – on enhancing Chat GPT's competence in effectively solving quadratic root functions. The integration of Artificial Intelligence (AI) into mathematical problem-solving has paved the way for innovative approaches. In this study, we delve into the Algebraic Formula Method and the Newton Sum Method, essential techniques for solving quadratic root functions. We aim to showcase the profound influence of these methods on Chat GPT's capacity to excel in solving quadratic equations. Through concrete evidence, we demonstrate Chat GPT's adept utilization of the Newton Sum Method for quadratic root function calculations. While Chat GPT can compute quadratic root functions of the form additionusing this method, its proficiency in using algebraic formula methods typically extends only up to addition . This marked discrepancy underscores the pivotal role that different methods play in amplifying the AI system's mathematical capabilities.  Chat GPT excels in solving quadratic equations using the Newton Sum Method compared to the Algebraic Formula Method. The model adeptly computes expressions of the form addition using this method, while its proficiency using algebraic formula methods is generally limited to addition . This striking discrepancy underscores the transformative impact that different methods can have on elevating the AI system's mathematical prowess. The research value is Pioneering Novel Maths Approaches for Overcoming Limitations in AI.  This study serves as an illuminating testament to the significance of pioneering innovative methodologies, rules, theorems, or formulas to surmount the current limitations in AI systems like Chat GPT. These innovative pursuits hold the key to unlocking the untapped potential that lies within, propelling AI systems to greater heights of proficiency. In essence, they offer a strategic pathway towards expanding the capabilities of AI and pushing the boundaries of what can be achieved.