Splitting of quadratic forms under some generic field extensions

Let $F$ be a field, $\operatorname{char}\!F\ne2$, $\varphi$ and $\psi$ be anisotropic quadratic forms over $F$. Let $L$ be the generic field extension of $F$ such that $i(\psi_L)\ge2$. Under which conditions is the form $\varphi_L$ isotropic? We give the answer to this question in the cases where $\dim\varphi=5$, $\dim\psi=6$ and $\dim\varphi=6$, $\dim\psi=7$.