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.

....(1) ....(2)

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

we have:

and

solving for and repectively:

....(I) and .....(II)

dividing (I) and (II):

....(3)

subtracting (I) and (II):

....(4)

because of another relation between roots and coefficients

we have:

.....(III) and ....(IV)

dividing (III) and (IV):

....(5)

subtracting (III) and (IV):

....(6)

From (5) and (3) :

.....(7)

From (4) and (6) :

......(8)

Finally, from (7) and (8):

If there are corrections or misunderstandings, feel free to reply, they are wellcome.