The example that you mention is the semidirect product for the multiplicative group $k^\times$ acting nontrivially on the additive group $k^+$. Its homology coincides with the homology of $k^\times$. To see this you can a Hochschild-Lyndon-Serre spectral sequence. The point is that $H_i(Bk^\times;H_j(Bk^+))=0$ when $j>0$ and $i\ge 0$. For this it suffices if $k^\times $ has a subgroup $T$ such that $H_i(BT;H_j(Bk^+))=0$ for all $i\ge 0$. Choose $T$ to be infinite cyclic, generated by a rational number $c\notin\lbrace 0,1,-1\rbrace$. $c$ acts on the rational vector space $H_j(Bk^+)$ by $c^j$, so that the homology of $BT$ is $H_0=coker(c^j-1)=0$, $H_1=ker(c^j-1)=0$, $H_i=0$ for $i>1$.

The example that you mention is the semidirect product for the multiplicative group $k^\times$ acting nontrivially on the additive group $k^+$. Its homology coincides with the homology of $k^\times$. To see this you can a Hochschild-Lyndon-Serre spectral sequence. The point is that $H_i(Bk^\times;H_j(Bk^+))=0$ when $j>0$ and $i\ge 0$. For this it suffices if $k^\times $ has a subgroup $T$ such that $H_i(BT;H_j(Bk^+))=0$ for all $i\ge 0$. Choose $T$ to be infinite cyclic, generated by a rational number $c\notin\lbrace 0,1,-1\rbrace$. $c$ acts on the rational vector space $H_j(Bk^+)$ by $c^{2j}$, so that the homology of $BT$ is $H_0=coker(c^{2j}-1)=0$, $H_1=ker(c^{2j}-1)=0$, $H_i=0$ for $i>1$.