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    Show that the curves y = x^4 − 2 and y = kx^2intersect for all values of k.
    i found the discriminant k2 + 8 > 0
    bur cant get the next mark?
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    (Original post by 321nassah)
    Show that the curves y = x^4 − 2 and y = kx^2intersect for all values of k.
    i found the discriminant k2 + 8 > 0
    bur cant get the next mark?
    If you let t=x^2 then you have the equation:

    t^2 - kt - 2 = 0

    The discriminant (as you said) is k^2+8 which is greater than 0 so the equation has solutions. But if both the solutions were negative then

    x^4 - kx^2 - 2 = 0

    would have no solutions.


    If you use the quadratic formula, the solutions are

    x^2 = (k +/- sqrt(k^2 + 8))/2

    How can you show that one of these solutions has to be positive?
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    i still don't get it
    does it have something to do with the x^2
    or does it have something to do with the discriminant
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    (Original post by 321nassah)
    i still don't get it
    does it have something to do with the x^2
    or does it have something to do with the discriminant
    Can you tell me the first line if my last post that you don't understand?

    It may help if you post your own working to show me how you arrived at k^2 + 8 >0.
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    i equated the 2 equations
    then used discriminant
    i know that k^2 is a positive interger + another postive interger still gives a postive interger, therefore its >0
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    (Original post by 321nassah)
    i equated the 2 equations
    then used discriminant
    i know that k^2 is a positive interger + another postive interger still gives a postive interger, therefore its >0
    Do you understand that saying the derterminant is > 0 is not enough to prove that the equation has solutions?

    And do you understand how I arrived at this:

    x^2 = (k +/- sqrt(k^2 + 8))/2

    ?


    It's hard to help you unless I know which parts you understand/don't understand. Please try to give a full explanation.
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    i understood how you arrived at those solution but just need help with the "How can you show that one of these solutions has to be positive?". i get everything else
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    (Original post by 321nassah)
    i understood how you arrived at those solution but just need help with the "How can you show that one of these solutions has to be positive?". i get everything else
    Oh sorry - from a previous post I assumed that you didn't understand everything.

    So the equation is:

    x^2 = (k +/- sqrt(k^2 + 8))/2

    We need to show that at least one of these solutions is positive. It makes sense to look at the one with '+' instead of '-':

    (k + sqrt(k^2 + 8))/2

    I claim that this is positive for all k and that's because sqrt(k^2 + 8) has to be greater than k.

    Can you see why that is?
 
 
 
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