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Math Help - Doubt in non rational equation

  1. #1
    Junior Member theprof's Avatar
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    Doubt in non rational equation

    Have a look at this irrational equation

    Its domain is [1;+infty).
    The main math packages (Derive, Maple and Mathematica) we have here, at school, give x = 0 and x = 2, both as solutions!
    Actually if you square twice, at last you get:
    x - 2x = 0
    that leads to x = 0 and x = 2, but x = 0 IMHO is not a solution...

    What do you think about?

    best regards
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  2. #2
    dr6583
    Guest
    The "domain" mentioned assumes real numbers only. Apparently the packages mentioned assumes complex numbers.

    When x=0:

    sqrt(-1) - sqrt(-1) = sqrt(0)
    i - i = 0
    0 = 0 ...so x=0 satisfies the equation.

    If you limit yourself to real numbers, then indeed x=0 is not a solution since sqrt(-1) is not a real number. If one squares both sides of the equation twice as mentioned, then solves the resulting quadratic, x=0 is considered an extraneous solution, unless complex solutions are assumed.

    Darrell
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  3. #3
    MHF Contributor
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    Yes, x=0 is not a solution because the given domain of x is from 1 to infinity only.
    Zero is not in the domain of x.
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  4. #4
    hpe
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    Mathcad (v.8 and 9) has the same funny behavior.

    Try this: Ask Maple to solve

    sqrt(5x-2) - sqrt(x-1) = sqrt(2x). (The only solution in the domain [1, infty) is x = 1/2 + sqrt(2)/2).

    Mathcad returns this solution plus the solution x = 1/2 - sqrt(2)/2 and believes that each solution has multiplicity four!
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