f(x) = ax2 + bx + c
f(1) = 0 ---> a(1)2 + b(1) + c = 0 ---> a + b + c = 0 (equation #1)
f(2) = 1 ---> a(2)2 + b(2) + c = 1 ---> 4a + 2b + c = 1 (#2)
f(3) = 8 ---> a(3)2 + b(3) + c = 8 ---> 9a + 3b + c = 8 (#3)
Subtracting #1 from #2: 4a + 2b + c = 1
- ( a + b + c = 0 )
---> 3a + b = 1 (#4)
Subtracting #2 from #3: 9a + 3b + c = 8
- ( 4a + 2b + c = 1 )
---> 5a + b = 7 (#5)
Subtracting #4 from #5: 5a + b = 7
- ( 3a + b = 1 )
---> 2a = 6
---> a = 3
Using #4: 3a + b = 1 ---> 3(3) + b = 1 ---> 9 + b = 1 ---> b = -8
Using #1: a + b + c = 0 ---> 3 + -8 + c = 0 ---> c = 5
Answer: f(x) = 3x2 - 8x + 5