Standard form equation of a vertical parabola:  

A (1, 1)
 ------ Substituting A (1, 1)
a + b + c = 1 --------- eq (i)

B (2, 2)
 ------ Substituting B (2, 2)
4a + 2b + c = 2 ------- eq (ii)

C (- 1, 5)
 --- Substituting C (- 1, 5)
a - b + c = 5 --------- eq (iii)

2b = - 4 ------- Subtracting eq (iii) from eq (i)


a - 2 + c = 1 ------ Substituting - 2 for b in eq (i) 
a + c = 3 ---------- eq (iv)

4a - 4 + c = 2 ----- Substituting - 2 for b in eq (ii)  
4a + c = 6 --------- eq (v)
3a = 3 ------------- Subtracting eq (iv) from eq (v)


1 - 2 + c = 1 ------ Substituting 1 for a, and - 2 for b in eq (i)
- 1 + c = 1


 
 ----- Substituting 1 for a, - 2 for b, and 2 for c