From the Player's Basic Rules v 0.2, page 35:
If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.
Generally, if something lets you wait until after the roll and see how you did before you use it, it specifies that. "When you make" a roll suggests that you use it at the same time as the roll, which isn't that helpful, but its effect is to give you advantage, which says (page 57):
Advantage and Disadvantage
Sometimes a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll. When that happens, you roll a second d20 when you make the roll.
This is more helpful, because it says you roll the second d20 when you make the roll. So advantage doesn't allow you to make the roll, then use your second d20; you roll them both at the same time. Since inspiration just gives advantage, it seems clear that you have to decide to use it before you roll.