Fraction and Decimal: Reciprocal of x's non-integer

The reciprocal part of $x$'s non-integer decimal part equals $x+1$, and $x>0$. What is $x$?

Solution: I tried this way-

Let's $n$= integer part of $x$

$1/x-n=x+1$

or, $1=(x-n)(x+1)$

or, $1={x}^{2}+x-nx-n$

or, ${x}^{2}+(1-n)x-(n+1)=0$

but, stucked here. Is there any other way?

The reciprocal part of $x$'s non-integer decimal part equals $x+1$, and $x>0$. What is $x$?

Solution: I tried this way-

Let's $n$= integer part of $x$

$1/x-n=x+1$

or, $1=(x-n)(x+1)$

or, $1={x}^{2}+x-nx-n$

or, ${x}^{2}+(1-n)x-(n+1)=0$

but, stucked here. Is there any other way?