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Homework Help: Cubic equation roots.

  1. Mar 7, 2010 #1
    1. The problem statement, all variables and given/known data
    The equation
    x3 − 3x2 + px + 4 = 0,
    where p is a constant, has roots α −β , α and α + β , where β > 0.
    (a) Find the values of α and β .
    (b) Find the value of p.
    how do i start off? all i know is that sigma a= -b/a and ab= c/a and ab(gamma) = -d/a .
    Would this be one of the ways to do it: [x-(a-β)] [x-(a+β)] (x-a).
    Thanks.


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 7, 2010 #2

    rock.freak667

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    You know that ∑α=-b/a, so use that to find α. ∑αβ=c/a and ∑αβγ=-d/a.

    So start off using the first method. Sum of all of the roots and equate it to 3. Do a similar exercise for the product of the roots.
     
  4. Mar 7, 2010 #3
    Oo is this how it works: ∑α=-b/a => -(-3)/1 = 3.
    to find b: ∑ab=c/a
    = ∑a.∑b= c/a.
    3∑b=px ... b=px/3. <--- sorry for such a weird presentation. Not sure :(.
     
  5. Mar 7, 2010 #4

    rock.freak667

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    Yes, but ∑α is the sum of the roots, your roots are α−β , α and α + β, what are the sum of the roots?
     
  6. Mar 7, 2010 #5
    3a?:confused:
     
  7. Mar 7, 2010 #6

    rock.freak667

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    Right yes good 3α.

    So ∑α=3α=3, what is α then?
     
  8. Mar 8, 2010 #7
    Oh! its one.. so to find out constant b would i mutiply out my roots:when a=1
    (1-b) (1+b) (1) = i think its wrong =/ .
     
  9. Mar 8, 2010 #8

    HallsofIvy

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    Another way to do this would be to set
    [tex](x- \alpha- \beta)(x- \alpha)(x- \alpha+ \beta)= x^2- 3x^2+ px+ 4[/tex]
    multiply it out and set corresponding coefficients equal. That gives you three equations for [itex]\alpha[/itex], [itex]\beta[/itex], and p.
     
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