# Fraction and Decimal: Reciprocal of x's non-integer The reciprocal part of x 's non-integer d

Lovellss 2022-06-25 Answered
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, $1=\left(x-n\right)\left(x+1\right)$
or, $1={x}^{2}+x-nx-n$
or, ${x}^{2}+\left(1-n\right)x-\left(n+1\right)=0$
but, stucked here. Is there any other way?
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## Answers (2)

Esteban Johnson
Answered 2022-06-26 Author has 15 answers
To complete Ashvin Swaminathan's answer:
$x=\frac{n-1±\sqrt{{n}^{2}+2n+5}}{2}$
Because $x>0$, we take
$x=\frac{n-1+\sqrt{{n}^{2}+2n+5}}{2}$
Since the discriminant ${n}^{2}+2n+5=\left(n+1{\right)}^{2}+4\ge 4$, then there are indeed infinitely many solutions, and these depend on $n=⌊x⌋$

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landdenaw
Answered 2022-06-27 Author has 8 answers
Solve for $x$ using the quadratic formula. As stated, there are infinitely many solutions...

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