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\[x=\frac{ 1 }{ 2 } + \frac{ 1 }{ 2 }\sqrt{5}\]

\[\frac{ x^{2}+x ^{-2} }{ x-x ^{-1} }=3\]

\[x=\frac{1}{2}+\frac{\sqrt{5}}{2}\]
\[x=\dfrac{1+\sqrt{5}}{2}\]

\[x^2+x^{-2}=3x-3x^{-1}\]
Multiply both sides by x^2.
\[x^4+1=3x^3-3x\]
\[x^4-3x^3+3x+1=0\]

i dont get it... both of you :P

yes ok

first, we find the value of \[x-\frac{1}{x}\]

You just sub in \[\frac{1+\sqrt{5}}{2}\]
into the original equation....

And you prove that the LHS=RHS.

but no matter how manytimes i try i can never get it to equal 3

Jeez, you have a tool that can help you but you don't want to use it.

You get 3 as the denominator when you plug the values into the numerator...

numerator*

we need to show our working our using our skill of indices laws... But i dont have much

There's no indice work when you learn that
\[x^{-1}=\frac{1}{x^1}\]
\[x^{-2}=\frac{1}{x^2}\]

I can solve that by hand but that will take me ages, to write it on the computer, so no thank you...

or this

thank you enka

you're welcome :)