What is the final speed of a car if it starts at 32 m/s and loses 1800J of energy?

I think the equation should be:
0.5m(32)^2 - 1800J = 0.5mv^2
where m = mass of the car
v = final speed.

Since there is only one equation with 2 unknown, I can only solve it to:

v = (1024 - 3600/m)^1/2

Is this correct?

yes. You need the mass of the card to solve for velocity.