This paper presents a comprehensive analytical investigation into solving a specific nonlinear Fredholm integral equation of the second kind, expressed as $u(x) = x + \lambda \int_0^1 x t u^2(t) \, dt$. Utilizing the Laplace-series method, we derive explicit solutions and validate their accuracy through detailed mathematical procedures. The study focuses on the parameter $\lambda$, with particular emphasis on the case $\lambda = 0.7$, where two distinct linear solutions emerge. We explore the derivation process, verify the solutions against special cases, and analyze their graphical representation using a MATLAB-based approach. The findings underscore the effectiveness of the Laplace-series method in addressing nonlinear integral equations and provide insights into the behavior of the solutions over the interval $[0,1]$. The results are further supported by numerical verification and a visual plot, offering a robust framework for understanding the equation’s solution space.