Beispiel Nr: 22
$ \text{Gegeben:} ax^{3}+bx^{2}+cx+d=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ \\ \textbf{Gegeben:} \\ -\frac{27}{28}x^3-\frac{27}{28}x^2+5\frac{11}{14}x =0\\ \\ \textbf{Rechnung:} \\ x(-\frac{27}{28}x^2-\frac{27}{28}x+5\frac{11}{14})=0 \Rightarrow x=0 \quad \vee \quad-\frac{27}{28}x^2-\frac{27}{28}x+5\frac{11}{14}=0\\ \\ -\frac{27}{28}x^{2}-\frac{27}{28}x+5\frac{11}{14} =0 \\ x_{1/2}=\displaystyle\frac{+\frac{27}{28} \pm\sqrt{\left(-\frac{27}{28}\right)^{2}-4\cdot \left(-\frac{27}{28}\right) \cdot 5\frac{11}{14}}}{2\cdot\left(-\frac{27}{28}\right)} \\ x_{1/2}=\displaystyle \frac{+\frac{27}{28} \pm\sqrt{23,2}}{-1\frac{13}{14}} \\ x_{1/2}=\displaystyle \frac{\frac{27}{28} \pm4\frac{23}{28}}{-1\frac{13}{14}} \\ x_{1}=\displaystyle \frac{\frac{27}{28} +4\frac{23}{28}}{-1\frac{13}{14}} \qquad x_{2}=\displaystyle \frac{\frac{27}{28} -4\frac{23}{28}}{-1\frac{13}{14}} \\ x_{1}=-3 \qquad x_{2}=2 \\ \underline{x_1=-3; \quad1\text{-fache Nullstelle}} \\\underline{x_2=0; \quad1\text{-fache Nullstelle}} \\\underline{x_3=2; \quad1\text{-fache Nullstelle}} \\$