Beispiel Nr: 38
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -2\frac{2}{9}x^2-2\frac{2}{9}x+4\frac{4}{9} =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -2\frac{2}{9}x^{2}-2\frac{2}{9}x+4\frac{4}{9} =0 \\ x_{1/2}=\displaystyle\frac{+2\frac{2}{9} \pm\sqrt{\left(-2\frac{2}{9}\right)^{2}-4\cdot \left(-2\frac{2}{9}\right) \cdot 4\frac{4}{9}}}{2\cdot\left(-2\frac{2}{9}\right)} \\ x_{1/2}=\displaystyle \frac{+2\frac{2}{9} \pm\sqrt{44\frac{4}{9}}}{-4\frac{4}{9}} \\ x_{1/2}=\displaystyle \frac{2\frac{2}{9} \pm6\frac{2}{3}}{-4\frac{4}{9}} \\ x_{1}=\displaystyle \frac{2\frac{2}{9} +6\frac{2}{3}}{-4\frac{4}{9}} \qquad x_{2}=\displaystyle \frac{2\frac{2}{9} -6\frac{2}{3}}{-4\frac{4}{9}} \\ x_{1}=-2 \qquad x_{2}=1 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -2\frac{2}{9}x^{2}-2\frac{2}{9}x+4\frac{4}{9} =0 \qquad /:-2\frac{2}{9} \\ x^{2}+1x-2 =0 \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm\sqrt{\left(\frac{1}{2}\right)^2- \left(-2\right)} \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm\sqrt{2\frac{1}{4}} \\ x_{1/2}=\displaystyle -\frac{1}{2}\pm1\frac{1}{2} \\ x_{1}=1 \qquad x_{2}=-2 \end{array}\\ \end{array}$