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Finding the interval of expression having two quadratic equations.

  1. Sep 2, 2011 #1
    1. The problem statement, all variables and given/known data

    What will be the values of 'm' so that the range of the equation
    [tex]y= \frac{mx^2+3x-4}{-4x^2+3x+m}[/tex]

    will be all real values i.e. [tex]y\epsilon (-\infty,\infty)[/tex]


    given:x can take all real values.
    any help or hint will be appreciated.

    2. Relevant equations



    3. The attempt at a solution
    i tried to find the range of the numerator and the denominator
    by using the formula
    if a<0 then range= (-infinity, -D/4a)

    if a>0 then range= (-D/4a, infinity)
    i couldn't proceed further.
    please help
    or provide hints.
     
    Last edited: Sep 2, 2011
  2. jcsd
  3. Sep 2, 2011 #2

    Dick

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    I have no idea what 'a' or 'D' are supposed to be. Here's a hint. What happens if the denominator is never equal to zero? For what values of m is the denominator never equal to zero?
     
  4. Sep 2, 2011 #3
    D= Discriminant
    a=coefficient of x^2
     
  5. Sep 2, 2011 #4

    Dick

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    Well ok, that's fine then. Again, think about what happens if the denominator is never equal to zero. That means the denominator is always the same sign. Can the range be [-infinity,infinity]? If so how?
     
    Last edited: Sep 2, 2011
  6. Sep 2, 2011 #5
    sorry the range should be(-infinity, infinity)
    i corrected the brackets [] -->()



    if the denominator is of same sign
    then to get all real values from the equation
    only the numerator will have to be considered
     
  7. Sep 3, 2011 #6

    Dick

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    Yes, so consider the numerator. If it has a lower or upper bound can the range be (-infinity,infinity)?
     
  8. Sep 3, 2011 #7
    lower or upper bound =?


    if denominator is not zero and coefficient of x^2 is negative then the whole denominator will be negative

    now as the denominator is negative
    if numerator is positive the whole equation will be negative
    and
    if numerator is negative the whole equation will be positive
     
    Last edited: Sep 3, 2011
  9. Sep 3, 2011 #8

    Dick

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    Try a concrete example. Suppose the range of the denominator is (-inf,-1/2] (so it's never zero) and the range of the numerator is [-1,inf). Can you show in that case that the range of the ratio isn't (-inf,inf)? Can you find a number that can't be in the range?
     
  10. Sep 3, 2011 #9

    eumyang

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    Last edited by a moderator: Apr 26, 2017
  11. Sep 3, 2011 #10
    no

    thank you very much.
     
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