1. (7p) Solve in the set of real numbers the equation:

$$
\{x\}^{2}+22\{x\}=10 x-9
$$

where $\{a\}$ denotes the fractional part of the real number $a$.

GM 12/2015
1. Let $[x]=a \in Z,\{x\}=b$ so $x=a+b$ and the equation becomes:

$b^{2}+12 b=10 a-9$

$b \in[0,1)$, so $0 \leq 10 a-9 \leq 13$, from which $a \in\{1,2\}$

For $a=1$ we get $b=\sqrt{37}-6$, so $x=\sqrt{37}-5$

For $a=2$ we get $b=\sqrt{47}-6$, so $x=\sqrt{47}-5$