Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

Let $Q$ be a nondegenerate quadratic form and $L$ a nonzero linear form of dimension $d>3$. As a generalization of the Oppenheim conjecture, we prove that the set $\{(Q(x),L(x)):x\in\mathbb{Z} ^d\}$ is dense in $\mathbb{R} ^2$ provided that $Q$ and $L$ satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.