The symmetry of the parabola dictates that if the vertex is at (5, 3) and it goes through (2, 0) then it must also go through (8, 0).
So there are three conditions from which to find the three unknowns:
3 = 25a + 5b + c. (1)
0 = 4a + 2b + c. (2)
0 = 64a + 8b + c. (3)
From (2) we get. c = -4a - 2b. (4)
Put this into (1) and (2).
3 = 21a + 3b. (5)
0 = 60a + 6b. (6)
Divide (5) by 3 and (6) by 6:
1 = 7a + b. (7)
0 = 10a + b. (8)
From (8) we get. b = -10a. (9)
Put this into (7)
1 = -3a. or: a = -1/3
Put this into (9): b = 10/3
Put these into (4): c = -16/3