Question

Could someplease explain the whole problem? For the first part my the answer I got was x= .618, -.618, and .z-dn.net/?f=%5Cfrac%7B%5Csqrt%5B1%2B-%5D%7B5%7D%20%7D%7B2%7D" id="TexFormula1" title="\frac{\sqrt[1+-]{5} }{2}" alt="\frac{\sqrt[1+-]{5} }{2}" align="absmiddle" class="latex-formula">

Answer 1

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Answer:

  a) x = (√5 -1)/2 ≈ 0.618034

  b) 1/x = (√5 +1)/2 ≈ 1.618034

Step-by-step explanation:

Given:

  x/(1 -x) = 1/x

Find:

  Exactly, and as a decimal approximation, ...

  a) x, using the quadratic formula

  b) 1/x

Solution:

a) We can multiply the given equation by x(1 -x) to obtain ...

  x² = 1 -x

  x² +x -1 = 0 . . . . . . add x-1

The coefficients for use in the quadratic formula are a=1, b=1, c=-1. The solution using the quadratic formula is ...

  $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}=\dfrac{-1\pm\sqrt{1^2-4\cdot1\cdot(-1)}}{2\cdot1}=\dfrac{-1\pm\sqrt{5}}{2}$

We are only interested in the positive solution, which is ...

  $\boxed{x=\dfrac{\sqrt{5}-1}{2}\approx0.618034}$

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b) The quadratic we developed in the first part can be rearranged like this:

  x² +x = 1 . . . . . . add 1 to both sides

  x(x +1) = 1 . . . . . factor out x

  x +1 = 1/x . . . . . .divide by x

Then to find the value of 1/x, we simply need to add 1 to the value of x we have:

  $\dfrac{1}{x}=\dfrac{\sqrt{5}-1}{2}+1=\dfrac{\sqrt{5}-1}{2}+\dfrac{2}{2}=\dfrac{\sqrt{5}-1+2}{2}\\\\\boxed{\dfrac{1}{x}=\dfrac{\sqrt{5}+1}{2}\approx1.618034}$