Beispiel Nr: 29
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -1\frac{1}{4}x^2+5x =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline -1\frac{1}{4}x^{2}+5x =0 \\ x(-1\frac{1}{4}x +5)=0 \\ \\ -1\frac{1}{4} x+5 =0 \qquad /-5 \\ -1\frac{1}{4} x= -5 \qquad /:\left(-1\frac{1}{4}\right) \\ x=\displaystyle\frac{-5}{-1\frac{1}{4}}\\ x_1=0\\ x_2=4 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -1\frac{1}{4}x^{2}+5x+0 =0 \\ x_{1/2}=\displaystyle\frac{-5 \pm\sqrt{5^{2}-4\cdot \left(-1\frac{1}{4}\right) \cdot 0}}{2\cdot\left(-1\frac{1}{4}\right)} \\ x_{1/2}=\displaystyle \frac{-5 \pm\sqrt{25}}{-2\frac{1}{2}} \\ x_{1/2}=\displaystyle \frac{-5 \pm5}{-2\frac{1}{2}} \\ x_{1}=\displaystyle \frac{-5 +5}{-2\frac{1}{2}} \qquad x_{2}=\displaystyle \frac{-5 -5}{-2\frac{1}{2}} \\ x_{1}=0 \qquad x_{2}=4 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -1\frac{1}{4}x^{2}+5x+0 =0 \qquad /:-1\frac{1}{4} \\ x^{2}-4x+0 =0 \\ x_{1/2}=\displaystyle -\frac{-4}{2}\pm\sqrt{\left(\frac{\left(-4\right)}{2}\right)^2- 0} \\ x_{1/2}=\displaystyle 2\pm\sqrt{4} \\ x_{1/2}=\displaystyle 2\pm2 \\ x_{1}=4 \qquad x_{2}=0 \end{array}\\ \end{array}$