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Quadratic Equations

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Let a,b,c be real numbers with a>0 such that the quadratic equation ##ax^2+bcx+b^3+c^3-4abc=0## has non real roots. Let ##P(x)=ax^2+bx+c## and ##Q(x)=ax^2+cx+b##. Which of the following is true?
    a) ##P(x)>0 \forall x \in R## and ##Q(x)<0 \forall x \in R##
    b) ##P(x)<0 \forall x \in R## and ##Q(x)>0 \forall x \in R##
    c) neither ##P(x)>0 \forall x \in R## nor ##Q(x)>0 \forall x \in R##
    d) exactly one of P(x) or Q(x) is positive for all real x.

    2. Relevant equations



    3. The attempt at a solution
    The first equation has non real roots which its discriminant is less than zero.
    [tex]b^2c^2-4a(b^3+c^3-4abc<0[/tex]
    [tex]\Rightarrow b^2c^2-4ab^3-4ac^3+16a^2bc<0[/tex]
    [tex]\Rightarrow b^2(c^2-4ab)-4ac(c^2-4ab)<0[/tex]
    [tex]\Rightarrow (b^2-4ac)(c^2-4ab)<0[/tex]

    ##b^2-4ac## is the discriminant of P(x) and ##c^2-4ab## is the discriminant for Q(x) and both the discriminants are less than which means both P(x) and Q(x) are greater than zero for all ##x \in R##.

    But there is no option which matches my conclusion.

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Feb 19, 2013 #2

    jbunniii

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    ##(b^2-4ac)(c^2-4ab)<0## means that one of the discriminants is negative, and the other is positive.
     
  4. Feb 19, 2013 #3
    Oh yes, thanks! :smile:

    This means that the answer is c?
     
  5. Feb 19, 2013 #4

    jbunniii

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    If one of the discriminants is positive, that means the corresponding quadratic has real roots, right? So it can't be c.
     
  6. Feb 19, 2013 #5
    Woops, I meant d, I switched the options in my mind. :redface:
     
  7. Feb 19, 2013 #6

    jbunniii

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    At least it wasn't an exam! :biggrin:
     
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