Beispiel Nr: 08
$\text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -\frac{1}{3}x^2+2x =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline -\frac{1}{3}x^{2}+2x =0 \\ x(-\frac{1}{3}x +2)=0 \\ \\ -\frac{1}{3} x+2 =0 \qquad /-2 \\ -\frac{1}{3} x= -2 \qquad /:\left(-\frac{1}{3}\right) \\ x=\displaystyle\frac{-2}{-\frac{1}{3}}\\ x_1=0\\ x_2=6 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -\frac{1}{3}x^{2}+2x+0 =0 \\ x_{1/2}=\displaystyle\frac{-2 \pm\sqrt{2^{2}-4\cdot \left(-\frac{1}{3}\right) \cdot 0}}{2\cdot\left(-\frac{1}{3}\right)} \\ x_{1/2}=\displaystyle \frac{-2 \pm\sqrt{4}}{-\frac{2}{3}} \\ x_{1/2}=\displaystyle \frac{-2 \pm2}{-\frac{2}{3}} \\ x_{1}=\displaystyle \frac{-2 +2}{-\frac{2}{3}} \qquad x_{2}=\displaystyle \frac{-2 -2}{-\frac{2}{3}} \\ x_{1}=0 \qquad x_{2}=6 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -\frac{1}{3}x^{2}+2x+0 =0 \qquad /:-\frac{1}{3} \\ x^{2}-6x+0 =0 \\ x_{1/2}=\displaystyle -\frac{-6}{2}\pm\sqrt{\left(\frac{\left(-6\right)}{2}\right)^2- 0} \\ x_{1/2}=\displaystyle 3\pm\sqrt{9} \\ x_{1/2}=\displaystyle 3\pm3 \\ x_{1}=6 \qquad x_{2}=0 \end{array}\\ \end{array}$