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Đáp án và giải thích các bước giải:

Điều kiện : `x>0,x\ne1`

`P={x\sqrt[x]-1}/{x-\sqrt[x]}-{x\sqrt[x]+1}/{x+\sqrt[x]}+{x+1}/{\sqrt[x]}` 

`P={(\sqrt[x]-1)(x+\sqrt[x]+1)}/{\sqrt[x](\sqrt[x]-1)}-{(\sqrt[x]+1)(x-\sqrt[x]+1)}/{\sqrt[x](\sqrt[x]+1)}+{x+1}/{\sqrt[x]}` 

`P={x+\sqrt[x]+1-x+\sqrt[x]-1+x+1}/{\sqrt[x]}`

`P={(\sqrt[x]+1)^2}/{\sqrt[x]}`

`b)`

`P=9/2⇔{(\sqrt[x]+1)^2}/{\sqrt[x]}=9/2`

`⇒` `2(\sqrt[x]+1)^2=9\sqrt[x]`

`⇔` `2x-5\sqrt[x]+2=0`

`⇔` `2x-4\sqrt[x]-\sqrt[x]+2=0`

`⇔` `2\sqrt[x](\sqrt[x]-2)-(\sqrt[x]-2)=0`

`⇔` `(\sqrt[x]-2)(2\sqrt[x]-1)=0`

`⇔` \(\left[ \begin{array}{l}\sqrt{x}=2\\\sqrt{x}=\dfrac12\end{array} \right.\) 

`⇔` \(\left[ \begin{array}{l}x=4\\x=\dfrac14\end{array}(TMĐK) \right.\) 

Vậy `S={4,1/4}`