Let $\alpha >1$ be irrational. Several authors studied the numbers $ {\ell ^m(\alpha )=\inf \{\, |y|:y\in \Lambda _m,\, y\ne 0\}}, $ where $m$ is a positive integer and $\Lambda _m$ denotes the set of all real numbers of the form $y=\epsilon _0\alpha ^n+\epsilon _1\alpha ^{n-1}+\cdots +\epsilon _{n-1}\alpha +\epsilon _n$ with restricted integer coefficients $|\epsilon _i|\le m$. The value of $\ell ^1(\alpha )$ was determined for many particular Pisot numbers and $\ell ^m(\alpha )$ for the golden number. In this paper the value of $\ell ^m(\alpha )$ is determined for irrational numbers $\alpha $, satisfying $\alpha ^2=a\alpha \pm 1$ with a positive integer $a$.