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Math Help - Finding Horizontal Intercepts

  1. #1
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    Finding Horizontal Intercepts

    I have the function x^4-2*x^2-1

    I have fount the dirivative
    4x^3-4x

    and all stationary points as the first part of the question asked but I am stuck on the second half it says :

    (b) By using the substitution u = x^2, find the horizontal intercepts.

    For the most part I dont really understand the question and have no idea were to start to solve it, Any ideas ?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Finding Horizontal Intercepts

    I'm assuming by horizontal intercepts, they mean roots. Using the suggested substitution, we may write:

    f(u)=u^2-2u-1=0

    Now, use the quadratic formula (or complete the square) to find the roots in terms of u, then back substitute for u to find the 4 roots in terms of x.
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    Re: Finding Horizontal Intercepts

    Yes its is the roots they are looking for and from graphing I know thay are -1.554 and 1.554, I can solve for the roots o the equation you suggested I just cant see how I am going to get back to the correct answer ?
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    Re: Finding Horizontal Intercepts

    OK, first of all, do NOT round off until the very end. Keep answers exact where possible.

    Now, as you were instructed by Mark, if you have let \displaystyle \begin{align*} u = x^2 \end{align*} then you end up with \displaystyle \begin{align*} u^2 - 2u - 1 = 0 \end{align*}. Solve this EXACTLY using the Quadratic Formula or Completing the Square. Then you will have the possible values for \displaystyle \begin{align*} u \end{align*}.

    Once you have the values for \displaystyle \begin{align*} u \end{align*}, remember that \displaystyle \begin{align*} u = x^2 \end{align*} and use this to find the values of \displaystyle \begin{align*} x \end{align*}.
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    Re: Finding Horizontal Intercepts

    Ok using the quadratic equation I come up with

    U= - 0.414213562 and U =  2.414213562

    so if U = X^2 then X=sqrt(u)

    and indeed Sqrt(2.414213562)=1.554

    but I cant take the square of the neg so how do I get the value for -1.554 ?
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    Re: Finding Horizontal Intercepts

    Quote Originally Posted by maca404 View Post
    Ok using the quadratic equation I come up with

    U= - 0.414213562 and U =  2.414213562

    so if U = X^2 then X=sqrt(u)

    and indeed Sqrt(2.414213562)=1.554

    but I cant take the square of the neg so how do I get the value for -1.554 ?
    Do NOT round off your answers. What are your answers for u in their EXACT surd form?
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  7. #7
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    Re: Finding Horizontal Intercepts

    You mean in the form x=1+sqrt(2) and x=1-sqrt*(2)
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    Re: Finding Horizontal Intercepts

    Quote Originally Posted by maca404 View Post
    You mean in the form x=1+sqrt(2) and x=1-sqrt*(2)
    Almost. Actually it's \displaystyle \begin{align*} u = 1 \pm \sqrt{2} \end{align*}, not x.

    Now since you know that \displaystyle \begin{align*} u = x^2 \end{align*}, that gives \displaystyle \begin{align*} x^2 = 1 \pm \sqrt{2} \end{align*}. How will you solve for x?
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