Study of the first-order transition in the spin-1 Blume-Capel model by using effective-field theory

► In this Letter we have studied the spin-1 Blume-Capel model by using effective-field theory (EFT). ► The first-order line is obtained for the first time. ► The model presents second and first-order phase transitions. ► We propose a functional to treat the first-order line. ► We discuss other alternative by using EFT to study first-order line.