Beispiel Nr: 37
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -\frac{4}{9}x^2+\frac{4}{9}x+\frac{8}{9} =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -\frac{4}{9}x^{2}+\frac{4}{9}x+\frac{8}{9} =0 \\ x_{1/2}=\displaystyle\frac{-\frac{4}{9} \pm\sqrt{\left(\frac{4}{9}\right)^{2}-4\cdot \left(-\frac{4}{9}\right) \cdot \frac{8}{9}}}{2\cdot\left(-\frac{4}{9}\right)} \\ x_{1/2}=\displaystyle \frac{-\frac{4}{9} \pm\sqrt{1\frac{7}{9}}}{-\frac{8}{9}} \\ x_{1/2}=\displaystyle \frac{-\frac{4}{9} \pm1\frac{1}{3}}{-\frac{8}{9}} \\ x_{1}=\displaystyle \frac{-\frac{4}{9} +1\frac{1}{3}}{-\frac{8}{9}} \qquad x_{2}=\displaystyle \frac{-\frac{4}{9} -1\frac{1}{3}}{-\frac{8}{9}} \\ x_{1}=-1 \qquad x_{2}=2 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -\frac{4}{9}x^{2}+\frac{4}{9}x+\frac{8}{9} =0 \qquad /:-\frac{4}{9} \\ x^{2}-1x-2 =0 \\ x_{1/2}=\displaystyle -\frac{-1}{2}\pm\sqrt{\left(\frac{\left(-1\right)}{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}=2 \qquad x_{2}=-1 \end{array}\\ \end{array}$