if x^2-2ra[r]x+r=0,r=1,2,3 are three quadratic equations of which each pair has exactly one root common then the number of solutions of the triplet (a1,a2,a3) is
Your notation is a bit strange. Do you mean , , and ?
By the quadratic formula, the solution to the first is , to the second , and to the third . Compare those in pairs so see how you can get one common root.