Let $\displaystyle f=ax^2+bx+c$ be a quadratic polynomial in $\displaystyle \mathbb{R}[x]$, with $a\ne 0$ and $\displaystyle \mathcal{R}_f$ be its range.
Denote $\displaystyle \mathcal{A}_{i+1}$ be the set of real numbers that lie in $\displaystyle \mathcal{R}_f$ get from finding roots of equation $f(x)=n_i$ where $n_i\in\mathcal{A}_i$, $\mathcal{A}_0=\mathbb{Z}\cap \mathcal{R}_f$.
Let $\displaystyle A=\mathbb{Z}\cup \left(\bigcup_{i=0}^{\infty} \mathcal{A}_i\right)$ and $B=\{x|x\ge 0,x\in A\}$. Is $B$ dense in $\mathbb{R^+}$ ?
