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3x^2 + 2x - k = 0, find 3α^2 - 2β in terms of k

  1. Mar 4, 2014 #1
    1. The problem statement, all variables and given/known data

    Let k be a constant. If α and β are the roots of the equation 3x^2 + 2x - k = 0, find the value of 3α^2 - 2β in terms of k.

    2. Relevant equations



    3. The attempt at a solution

    Obviously, the usual

    αβ = -k/3
    α + β = -2/3

    has been written but I couldn't put them into the equation required despite a full hour's effort.

    Also, tried writing (-b+-sqrt(b^2-4ac))/2a, and put respective roots into the equation, it yields something similar to the provided solution, but has an extra root term.

    The solution is 4/3 + k.

    The latter method gets 4/3 + k + sqrt(4+12k)/6.
     
  2. jcsd
  3. Mar 4, 2014 #2

    Mark44

    Staff: Mentor

    Considering k to be a constant, solve the second equation below for α or β, then substitute into the first equation.
    αβ = -k/3
    α + β = -2/3
     
  4. Mar 4, 2014 #3
    It has a β^2 term leftover.
     
  5. Mar 4, 2014 #4

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Show your work in detail.

    ehild
     
  6. Mar 4, 2014 #5
    haha omg i got it
     
  7. Mar 4, 2014 #6
    got it

    i tried again this time giving each equation their names, suddenly spot the identity that i didn't see before. hahahaha
     

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