I came to a proof yesterday. I post my conclusion below(despite not being a difficult problem to be highlighted) so perhaps someone interested in the problem can see it.
If the two equations above have just one solution in common,
then. Let be: the roots of (1), and the roots of (2).
because of the relation between roots and coefficients of a quadratic equation
solving for and repectively:
....(I) and .....(II)
dividing (I) and (II):
subtracting (I) and (II):
because of another relation between roots and coefficients
.....(III) and ....(IV)
dividing (III) and (IV):
subtracting (III) and (IV):
From (5) and (3) :
From (4) and (6) :
Finally, from (7) and (8):
If there are corrections or misunderstandings, feel free to reply, they are wellcome.