The Ramanujan-Nagell equation x^2 + 7 = 2n is a classic Diophantine equation with integer solutions. In this paper, we explore this equation using the theory of elliptic curves and computational techniques. We transform the Ramanujan-Nagell equation into elliptic curves of the forms y^2 = x^3 − 7, y^2 = 2x^3 − 7, and y^2 = 4x^3 − 7, and use the Nagell-Lutz theorem to analyze torsion points and solutions. Our primary research question is: Can we classify all integer solutions to the Ramanujan-Nagell equation using elliptic curves and computational methods? We provide an expository presentation on the groups of rational points on elliptic curves, offering insights into their structure and properties. Computational methods using SageMath are employed to verify and explore solutions. Our work reproduces and classifies all integer solutions (2, 1) and (2, −1) for n = 3, (2, 3) and (2, −3) for n = 4, (2, 5) and (2, −5) for n = 5, (4, 11) and (4, −11) for n = 7, and (32, 181) and (32, −181) for n = 15, which are all solutions of the Ramanujan-Nagell equation. This analysis demonstrates the effectiveness of elliptic curves and computational techniques in solving Diophantine equations.