Algebra-Gleichungen-Quadratische Gleichung

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