Beispiel Nr: 28
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -\frac{32}{81}x^2-\frac{32}{81}x+7\frac{73}{81} =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -\frac{32}{81}x^{2}-\frac{32}{81}x+7\frac{73}{81} =0 \\ x_{1/2}=\displaystyle\frac{+\frac{32}{81} \pm\sqrt{\left(-\frac{32}{81}\right)^{2}-4\cdot \left(-\frac{32}{81}\right) \cdot 7\frac{73}{81}}}{2\cdot\left(-\frac{32}{81}\right)} \\ x_{1/2}=\displaystyle \frac{+\frac{32}{81} \pm\sqrt{12\frac{52}{81}}}{-\frac{64}{81}} \\ x_{1/2}=\displaystyle \frac{\frac{32}{81} \pm3\frac{5}{9}}{-\frac{64}{81}} \\ x_{1}=\displaystyle \frac{\frac{32}{81} +3\frac{5}{9}}{-\frac{64}{81}} \qquad x_{2}=\displaystyle \frac{\frac{32}{81} -3\frac{5}{9}}{-\frac{64}{81}} \\ x_{1}=-5 \qquad x_{2}=4 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -\frac{32}{81}x^{2}-\frac{32}{81}x+7\frac{73}{81} =0 \qquad /:-\frac{32}{81} \\ x^{2}+1x-20 =0 \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm\sqrt{\left(\frac{1}{2}\right)^2- \left(-20\right)} \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm\sqrt{20\frac{1}{4}} \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm4\frac{1}{2} \\ x_{1}=4 \qquad x_{2}=-5 \end{array}\\ \end{array}$