Solving a Maple Mystery: Why Does x=root(x^2+1)/x=.5root(2+root(20))?

[fredrick08]

Homework Statement

can someone please tell me why maple says that x=root(x^2+1)/x=.5root(2+root(20))??

ive tried trig and rearranging, but i have idea why this is so..

 

[danago]

Have you got a value stored into the variable x?

EDIT: Oh i think i may have misinterpreted the problem.

Heres some working:
http://img4.imageshack.us/img4/8552/algebra.gif

Probably is a quicker way, but these are the steps i took.

 

[sylas]

Homework Statement

can someone please tell me why maple says that x=root(x^2+1)/x=.5root(2+root(20))??

ive tried trig and rearranging, but i have idea why this is so..

Square both sides, and you get a quadratic in x². Solve.

$$\begin{align*} x &= \frac{\sqrt{x^2+1}}{x} \\ \intertext{Hence} x^2 &= \sqrt{x^2 + 1} \\ x^4 &= x^2 + 1 \\ x^2 &= \frac{1 \pm \sqrt{5}}{2} \\ \intertext{(Must be positive, so dismiss the negative part)} & = \frac{2 + \sqrt{20}}{4} \\ \intertext{Hence} x & = \pm 0.5 \sqrt{2 + \sqrt{20}} \\ &= \pm \sqrt{0.5 + \sqrt{1.25}} \end{align*}$$​

 

[fredrick08]

thankyou very much, I am still unsure how you get from x^4 to x^2=2+root(20)/4.. but i know its right so thx

 

[Cyosis]

He uses a substitution p=x^2 and then solves p with the ABC formula.

 

[fredrick08]

ok yes thankyou