The Solar X-ray Imager (SXI) is a complimentary, add-on instrument designed for use on the next generation of Geostationary Orbiting Environmental Satellites (GOES). Its primary mission is to continuously observe the full solar disc at X-ray wavelengths; including coronal holes, active regions, flares, and coronal mass ejections. For wide-field imaging applications, there is little merit in an optical design exhibiting stigmatic imaging on-axis; we therefore departed from the classical Wolter Type I design in favor of a hyperboloid-hyperboloid design that balances not only defocus with field curvature but also third-order spherical aberration and astigmatism with oblique spherical aberration. A detailed system engineering analysis including the substantial surface scattering and detector effects indicates that the resulting hyperboloid-hyperboloid design will achieve an 80% increase (over the baseline design) in the number of spatial resolution elements (and hence in total information content in the image) over an 18 arc min radius field-of-view. A comparison of performance predictions with X-ray test data for the SXI Engineering Model is included.