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Finding the unknowns of irrational equation

  1. Oct 3, 2007 #1
    1. The problem statement, all variables and given/known data
    For the following irrational equation

    [tex]x^2 + 7x + 10 + \sqrt{x^2 + 7x + 12} = 0[/tex]

    Find all possible unknown of X.

    2. Relevant equations

    None. Just your ability to solve equations.

    3. The attempt at a solution

    First of all, I am not allowed to use a calculator to solve this (During an exam).

    Second of all, I am not sure why this is considered as irrational equation, but I went ahead and attempted to solve it.

    [tex]x^2+7x+10 = -\sqrt{x^2+7x+12}[/tex]
    [tex]x^2+7x+10 = -\sqrt{(x+4)(x+3)}[/tex]
    [tex](x^2+7x+10)^2 = (x+4)(x+3)[/tex]
    [tex]x^4+14x^3+140x + 69x^2 + 100 = x^2 + 7x + 12[/tex]
    [tex]x^4 + 14x^3 + 68x^2 + 133x + 88 = 0[/tex]

    And I am completely stuck here. I know no method which you can solve this fourth order equation by hand (Remember, NO CALCULATORS TO SOLVE THIS)

    Any ideas?
  2. jcsd
  3. Oct 3, 2007 #2


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    The only thing you can try to do is to factor. And it only can be factored into two quadratics. And it would take a considerable amount of trial and error to find that. Do you want me to say that it's a really poor exam question? Because I would agree with that.
  4. Oct 3, 2007 #3


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    I don't know any way to do it either. I wonder if there's a misprint; if the 10 was a 12, it would be much easier ...
  5. Oct 4, 2007 #4
    Unfortunately, it actually is an exam question (Supposedly solvable). This was a question from Korean National Standarized Test. From what I heard, there's a real simpler way of doing this, but I just cannot figure out what it is.

    Last edited: Oct 4, 2007
  6. Oct 4, 2007 #5
    And yes, this can be solved very easily.

    Set [tex]u = x^2 + 7x + 10[/tex]

    [tex]u + \sqrt{u + 2} = 0[/tex]

    [tex]\sqrt{u + 2} = -u[/tex]

    [tex]u + 2 = u^2 [/tex]

    [tex]u^2 - u - 2 = 0[/tex]

    u = 2, -1

    Now set the each corresponding u to two different equations

    [tex]x^2 + 7x + 10 = 2[/tex]

    [tex]x^2 + 7x + 10 = -1[/tex]

    Solve for x.
  7. Oct 4, 2007 #6
    You are a freaking GENIUS! Thanks a lot!!!
  8. Oct 4, 2007 #7


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  9. Oct 4, 2007 #8


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    Dearly Missed

    Remember to get rid of false solutions!

    u=2 cannot be used, since
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