Puzzling solution for $1/x = 3 - 2\sqrt{x}$ What is the solution for
$$\frac{1}{x} = 3 - 2\sqrt{x}$$
When I plot $1/x$ and $3-2\sqrt{x}$ separately, it meets at $x = 1$.
However when I solve the equation $1/x = 3 - 2\sqrt{x}$ algebraically, I get two solutions $-1/4$ and $1$.
What am I missing?

Rewrite the equation as
$$
2\sqrt{x}=3-\frac{1}{x}
$$
This shows that there is an implied condition, besides $x>0$ necessary in order that the equation makes sense: indeed you need
$$
3-\frac{1}{x}\ge0
$$
which becomes, taking into account that $x>0$, $3x-1\ge0$, so $x\ge1/3$.
Now you can safely square and get
$$
4x=9-\frac{6}{x}+\frac{1}{x^2}
$$
and therefore
$$
4x^3-9x^2+6x-1=0
$$
This factors as $(x-1)^2(4x-1)=0$. Since $1/4<1/3$, this is a spurious solution that must be discarded. Hence the only solution is $x=1$.
