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Prove Existence Unique Real Solution

  1. Mar 22, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove Existence Unique Real Solution to
    ## x^{3} + x^{2} -1 =0 ## between ## x= \frac{2}{3} \text{and} x=1##


    3. The attempt at a solution

    ## x^{2} ( x+1) =1 ##
    I know that the solution is x =0.75488, but this came from some website. How do I find this number without a calculator?
     
  2. jcsd
  3. Mar 22, 2014 #2

    micromass

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    You don't actually need to find the solution. You just need to know it exists. Think intermediate value theorem.
     
  4. Mar 23, 2014 #3

    HallsofIvy

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    The intermediate value theorem shows that there exist a solution betweem 2/3 and 1. The mean value theorem shows that there is only one.
     
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