Preheat the oven to 375°F(220°C) conventional or 350°F(175°C) convection.
In a bowl, mix together the flour, cornmeal, baking powder, baking soda, salt, and sugar. Set aside.
In a separate bowl, whisk together the buttermilk, and slightly beaten eggs. Add discard if using.
Combine the wet mixture with the dry ingredients and stir. The batter should be lumpy.
Fold in melted butter.
Spread the batter evenly into a lightly greased 8-inch baking dish.
Bake for 20-25 minutes conventional or around 15-20 minutes convection until golden brown. Use a toothpick to test the center of the cornbread; it should come out clean.
Allow it to cool for about 15 minutes before serving.