2. Let planar vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy $|\boldsymbol{a}+\boldsymbol{b}|=3$, then the maximum value of $\boldsymbol{a} \cdot \boldsymbol{b}$ is $\qquad$
$\boldsymbol{a} \cdot \boldsymbol{b}=\frac{1}{4}\left[(\boldsymbol{a}+\boldsymbol{b})^{2}-(\boldsymbol{a}-\boldsymbol{b})^{2}\right] \leqslant \frac{9}{4}$, equality holds when $\boldsymbol{a}=\boldsymbol{b}$.