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Math Help - Real Solutions,

  1. #1
    Junior Member
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    Real Solutions,

    Hi,

    The graph y=\frac{2}{x^2}+x^4 has four real number solutions for values of y > 3.

    I can draw the graph using my math tools, but this is quite a stinker to try and graph by hand.

    I was wondering how you would find the four real solutions for a particular value of y, say 5. I have tried factoring it.

    \frac{2}{x^2}+x^4=5
    x^2(x^4-5)=2

    There doesn't appear to be any way of factoring out the four real number solutions. What is the correct way to find values of the graph.

    Thanks
    Regards
    Craig.
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  2. #2
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    Quote Originally Posted by craigmain View Post
    Hi,

    The graph y=\frac{2}{x^2}+x^4 has four real number solutions for values of y > 3.

    I can draw the graph using my math tools, but this is quite a stinker to try and graph by hand.

    I was wondering how you would find the four real solutions for a particular value of y, say 5. I have tried factoring it.

    \frac{2}{x^2}+x^4=5
    x^2(x^4-5)=2

    There doesn't appear to be any way of factoring out the four real number solutions. What is the correct way to find values of the graph.

    Thanks
    Regards
    Craig.
    \frac{2}{x^2}+x^4=5~\implies~x^6-5x^2+2=0

    If you use the substitution x^2 = y you'll get a reduced cubic equation:

    y^3-5y+2=0

    You now can apply the Cardanic formula to solve this equation. Don't forget to re-substitute to calculate the value of x.
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  3. #3
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    The fact that an equation has 4 solutions does not mean that those solutions are rational numbers or that they can be found easily!

    In this case, multiplying the left side, x^6- 5x^2= 2 or x^6- 5x^2- 2= 0. If you let y= x^2 you can write that as y^4- 5y- 2= 0, a cubic equation. By the "rational root" theorem, the only possible rational roots are \pm 1 or \pm 2, and it is easy to check that none of those satisfies the equation- thus the original equation has no rational roots.

    You could use Cardan's cubic formula
    Cubic function - Wikipedia, the free encyclopedia
    to solve the cubic, then take the square roots of that.
    Last edited by mr fantastic; March 28th 2009 at 03:54 PM. Reason: Fixed latex tags: Replaced [itex] with [math] etc.
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