Thread: A strange property of the Quadratic ecuation

I´ve found a strange and difficult to prove formula that derived from two quadratic equations.

"If two quadratic equations: and share one solution.
then: .......(1) is always true."

If they are equivalent, It is known that their coefficients are proportional: .
since the two quadratic equations above share just one solution, I suppose they are partially equivalent.

I've been trying to prove the formula in (1). I apologize for the lack of content about the problem.
Any suggestion will be appreciated.

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.