TUYMAADA 2024 J7

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that $$\frac{f(k)}{g(k)}=\frac{n + 1}{n}.$$ Prove that $f$ and $g$ have a common root. Proposed by A. Golovanov