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Homework Help: Help with finding all real solutions

  1. Sep 24, 2009 #1


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    1. The problem statement, all variables and given/known data
    x^4 - 8x^2 + 2 = 0

    2. Relevant equations
    quadratic formula?

    3. The attempt at a solution
    i would've tried using the quadratic formula, but im not sure if this would work with that seeing as how it's not ax^2 + bx + c
  2. jcsd
  3. Sep 24, 2009 #2
    You will need to use the factor/remainder theorem.
  4. Sep 24, 2009 #3


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    The example equation is in quadratic form, or something like it. First, you may use a substitution, something like, t=x^2. This can give you like, this:

    t^2 - 8t + 2 = 0

    Now, you can solve THAT one and you may obtain two solutions, but those solutions are for t. Now, solve each of those solutions for x (remember to first replace t with x^2 )
  5. Sep 24, 2009 #4
    The polynomial fortunately is quadratic, but in x2.

    If u = x2 then you have x4 -8x2 + 2 = u2 - 8u + 2 = 0.

    You can use the quadratic formula to solve for x2 keeping in mind that any negative roots from the formula must be discarded since x2 can never be negative.

  6. Sep 24, 2009 #5


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    We have this field now called complex numbers that has negitive squares.
    i^2=-1 for example
  7. Sep 25, 2009 #6
    The thread title is "help with finding all real solutions" so I figured that complex solutions weren't needed.

  8. Sep 25, 2009 #7


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    Good point. I like to keep track of the complex roots to assist in tracking the real roots. The fundamental theorem of a algebra impies each complex root means one less real root to find.
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