Order the steps that recover the position of a body from its constant acceleration.

1. Start with the constant acceleration a
2. Integrate a with respect to time to get v = at + C
3. Fix the constant using the initial velocity: v = v0 + at
4. Integrate v with respect to time to get x = v0 t + one half a t squared + C
5. Fix the second constant using the initial position: x = x0 + v0 t + one half a t squared

Going from acceleration to position is two integrations, and each one produces a constant that only the initial conditions can fix. This is why the equations of motion need both $v_0$ and $x_0$: two integrations, two constants.