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Help finding solutions to sqrt(a+sqrt(-b))+sqrt(a-sqrt(-b))=4 and

  1. Jul 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Q A
    Find all integers which satisfy sqrt(a+sqrt(-b))+sqrt(a-sqrt(-b))=4
    Q B
    Find all the roots for the quadratic equation x^2-x-(2+4+6+...+2014)


    2. Relevant equations

    ---

    3. The attempt at a solution
    Well with both I keep coming up with 0=0 and cannot seem to get past that
     
  2. jcsd
  3. Jul 1, 2013 #2

    Simon Bridge

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    How are you attempting the problems?
     
  4. Jul 1, 2013 #3
    In the first I tried simplifying and got a=8, but sqrt(-b) always cancelled out. For the second I tried substituting for x or x^2 and it always cancelled all the other values out.
     
  5. Jul 1, 2013 #4
    Well, for the second one, what are you getting for the summation of 2 to 2014?
    The roots I get for your second problem are x = 1008 and x = -1007

    I get 1,015,056 for the summation of positive integers from 2 to 2014.
    Then the problem becomes nothing more than factoring this:
    x² - x - 1,015,056
     
  6. Jul 1, 2013 #5

    Simon Bridge

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    You seem to be reluctant to show how you are thinking about the problems ... you say "I tried simplifying..." but don't tell us the simplification, you say "I tried substituting..." but do not tell us the substitution.

    For the second one - there is a formula for finding the roots of any quadratic.... why not use it?
    (hint: put c=that long sum on the end and find the roots wrt c).

    For the first one - you appear t be saying that you get a=8 and b can be anything... did you check this by putting a=8 and b=0 into the equation? What happened?

    ... may should try exploring the details a bit to get a feel for how it behaves.
    i.e. the relation could be thought of as x + y = 4.
    if x and y are real numbers, and x=y, what does that mean for a and b?
    what happens is x and y are complex numbers and x=y?
    You should find these situations don't cancel out... are there any more?
    Does a systematic way of finding the solutions occur to you?

    Overall - these problems must be set in the context of some lessons: what were the lessons about?
     
    Last edited: Jul 2, 2013
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