Beispiel Nr: 45
$\text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ \frac{8}{27}x^2+2\frac{2}{3}x =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline \frac{8}{27}x^{2}+2\frac{2}{3}x =0 \\ x(\frac{8}{27}x +2\frac{2}{3})=0 \\ \\ \frac{8}{27} x+2\frac{2}{3} =0 \qquad /-2\frac{2}{3} \\ \frac{8}{27} x= -2\frac{2}{3} \qquad /:\frac{8}{27} \\ x=\displaystyle\frac{-2\frac{2}{3}}{\frac{8}{27}}\\ x_1=0\\ x_2=-9 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ \frac{8}{27}x^{2}+2\frac{2}{3}x+0 =0 \\ x_{1/2}=\displaystyle\frac{-2\frac{2}{3} \pm\sqrt{\left(2\frac{2}{3}\right)^{2}-4\cdot \frac{8}{27} \cdot 0}}{2\cdot\frac{8}{27}} \\ x_{1/2}=\displaystyle \frac{-2\frac{2}{3} \pm\sqrt{7\frac{1}{9}}}{\frac{16}{27}} \\ x_{1/2}=\displaystyle \frac{-2\frac{2}{3} \pm2\frac{2}{3}}{\frac{16}{27}} \\ x_{1}=\displaystyle \frac{-2\frac{2}{3} +2\frac{2}{3}}{\frac{16}{27}} \qquad x_{2}=\displaystyle \frac{-2\frac{2}{3} -2\frac{2}{3}}{\frac{16}{27}} \\ x_{1}=0 \qquad x_{2}=-9 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ \frac{8}{27}x^{2}+2\frac{2}{3}x+0 =0 \qquad /:\frac{8}{27} \\ x^{2}+9x+0 =0 \\ x_{1/2}=\displaystyle -\frac{9}{2}\pm\sqrt{\left(\frac{9}{2}\right)^2- 0} \\ x_{1/2}=\displaystyle -4\frac{1}{2}\pm\sqrt{20\frac{1}{4}} \\ x_{1/2}=\displaystyle -4\frac{1}{2}\pm4\frac{1}{2} \\ x_{1}=0 \qquad x_{2}=-9 \end{array}\\ \end{array}$