\[\left( 3\frac{1}{4} \right)^{2} - \left( 4\frac{1}{2} \right)^{2} = \left( \frac{13}{4} \right)^{2} - \left( \frac{9}{2} \right)^{2} =\]
\[= \left( \frac{13}{4} \right)^{2} - \left( \frac{18}{4} \right)^{2} =\]
\[= \left( \frac{13}{4} - \frac{18}{4} \right)\left( \frac{13}{4} + \frac{18}{4} \right) =\]
\[= - \frac{5}{4} \cdot \frac{31}{4} = - \frac{155}{16} = - 9,6875.\]