Beispiel Nr: 41
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ \frac{5}{9}x^2-3\frac{1}{3}x =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline \frac{5}{9}x^{2}-3\frac{1}{3}x =0 \\ x(\frac{5}{9}x -3\frac{1}{3})=0 \\ \\ \frac{5}{9} x-3\frac{1}{3} =0 \qquad /+3\frac{1}{3} \\ \frac{5}{9} x= 3\frac{1}{3} \qquad /:\frac{5}{9} \\ x=\displaystyle\frac{3\frac{1}{3}}{\frac{5}{9}}\\ x_1=0\\ x_2=6 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ \frac{5}{9}x^{2}-3\frac{1}{3}x+0 =0 \\ x_{1/2}=\displaystyle\frac{+3\frac{1}{3} \pm\sqrt{\left(-3\frac{1}{3}\right)^{2}-4\cdot \frac{5}{9} \cdot 0}}{2\cdot\frac{5}{9}} \\ x_{1/2}=\displaystyle \frac{+3\frac{1}{3} \pm\sqrt{11\frac{1}{9}}}{1\frac{1}{9}} \\ x_{1/2}=\displaystyle \frac{3\frac{1}{3} \pm3\frac{1}{3}}{1\frac{1}{9}} \\ x_{1}=\displaystyle \frac{3\frac{1}{3} +3\frac{1}{3}}{1\frac{1}{9}} \qquad x_{2}=\displaystyle \frac{3\frac{1}{3} -3\frac{1}{3}}{1\frac{1}{9}} \\ x_{1}=6 \qquad x_{2}=0 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ \frac{5}{9}x^{2}-3\frac{1}{3}x+0 =0 \qquad /:\frac{5}{9} \\ 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}$