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Math Help - A strange property of the Quadratic ecuation

  1. #1
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    A strange property of the Quadratic ecuation

    Ive found a strange and difficult to prove formula that derived from two quadratic equations.

    "If two quadratic equations: Ax^2+Bx+C=0 and Dx^2+Ex+F=0 share one solution.
    then: (CD-FA)^2=(BF-EC)(AE-BD).......(1) is always true."

    If they are equivalent, It is known that their coefficients are proportional: \frac{A}{D}=\frac{B}{E}=\frac{C}{F}.
    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.
    Last edited by rochosh; May 13th 2012 at 04:28 PM.
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  2. #2
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    Re: A strange property of the Quadratic ecuation

    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.

    Ax^2+Bx+C=0....(1) Dx^2+Ex+F=0....(2)

    If the two equations above have just one solution in common,
    then. Let be: {r_1 , r} the roots of (1), and {r_2 , r} the roots of (2).

    because of the relation between roots and coefficients of a quadratic equation
    we have:

    r_1 + r =-B/A  and   r_2 + r =-E/D

    solving for r_1 and r_2 repectively:

    r_1 =\frac{-B-Ar}{A}  ....(I) and   r_2 =\frac{-E-Dr}{D}.....(II)

    dividing (I) and (II):

    \frac{r_1}{ r_2}=\frac{D(B+Ar)}{A(E+Dr)}....(3)

    subtracting (I) and (II):

    r_1-r_2=\frac{AE-DB}{AD}....(4)


    because of another relation between roots and coefficients
    we have:

    r_1 =C/Ar  .....(III) and   r_2 =F/Dr....(IV)

    dividing (III) and (IV):

    \frac{r_1}{r_2}= \frac{CD}{FA}....(5)

    subtracting (III) and (IV):

    r_1-r_2=\frac{CD-FA}{ADr}....(6)

    From (5) and (3) :

    r=\frac{FB-CE}{CD-FA}.....(7)

    From (4) and (6) :

    r=\frac{CD-FA}{AE-DB}......(8)


    Finally, from (7) and (8):

    (CD-FA)^2=(AE-DB)(FB-CE)

    If there are corrections or misunderstandings, feel free to reply, they are wellcome.
    Last edited by rochosh; May 17th 2012 at 06:21 PM.
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