3D Gaussian Splatting (3DGS) has emerged as an advanced technique for real-time novel view synthesis by representing scene geometry and appearance using differentiable Gaussian primitives. However, efficiently computing precise Gaussian-tile intersections remains a critical task in the rasterization pipeline. To this end, we propose QuadBox, a method that leverages four axis-aligned bounding boxes to tightly encapsulate projected Gaussians in a discrete manner. First, we derive a geometry-aware stretching factor that enables the construction of a tile-aligned QuadBox, which covers the elliptical projection and largely excludes irrelevant tiles. Second, we introduce QPass, a single-pass tile traversal algorithm that exhaustively exploits the discrete nature of QuadBox, ensuring that the tile intersection check is performed with simple interval tests. Experiments on public datasets show that our method accelerates the rendering speed of 3DGS by 1.85$\times$. Code is available at \href{https://github.com/Powertony102/QuadBox}{https://github.com/Powertony102/QuadBox}.