Let \({\mathbb{F}}_ q\) be the finite field with q elements, V be a finite- dimensional vector space over \({\mathbb{F}}_ q\), dim V\(=n\), \(L_ 1\), \(L_ 2\) linear forms and Q a quadratic form on V. The author proves an upper bound for the Kloosterman sum \[ K(L_ 1,L_ 2;Q):=\sum_{Q(v)\neq 0}\chi ((L_ 1(v)+L_ 2(v)(Q(v))^{-1}), \] where \(\chi\) is a non- trivial additive character on \({\mathbb{F}}_ q\). Suppose \(\ker (L_ 1)\neq \ker (L_ 2)\) and Q be nondegenerate. If \(n\equiv 0 (mod 2)\) assume in addition, that \(Q|_{\ker (L_ 1)}\) and \(Q|_{\ker (L_ 2)}\) are nondegenerate. The main result of the author then reads \[ | K(L_ 1,L_ 2;Q)| \leq 5q^{n/2}\text{ for } n\quad even\text{ and } \leq 4q^{n/2}\text{ for } n\quad odd. \] For the proof A. Weil's famous upper bound for Kloosterman sums is used. The remaining calculations stay elementary, yet most of them are not performed explicitly. His results should be compared with the work of \textit{F. Grunewald}, \textit{J. Elstrodt} and \textit{J. Mennicke} [C. R. Acad. Sci., Paris, Sér. I 305, 577-581 (1987; Zbl 0633.10025)] and \textit{N. Katz} and \textit{G. Laumon} [Publ. Math., Inst. Hautes Étud. Sci. 62, 145-202 (1985; Zbl 0603.14015)].