Given a+b+c=6, abc=6 and a^2+b^2+c^2=14

 (a + b+ c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
 6^2 = 14 + 2(ab + bc + ca)
 (36 - 14) / 2 = ab + bc + ca
 6c = acb + bc^2 + ac^2
 6c = 6 + c^2(a + b)
 6c = 6 + c^2(a + b + c - c)
 6c = 6 + c^2(6 - c)
 6c = 6 + 6c^2 - c^3

 c^3 - 6c^2 + 6c - 6 = 0

 c = 1

a + b + 1 = 6
a + b = 5

a * b * 1 = 6
a = 6 / b

6/b + b = 5
6 + b^2 = 5b
b^2 - 5b + 6 = 0

b = (2, 3)
a = (3, 2)
c = 1