Beispiel Nr: 27
$ \text{Gegeben:} ax^{2}+bx+c=0 \\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\ \\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\ -\frac{8}{49}x^2-\frac{24}{49}x+1\frac{31}{49} =0 \\ \\ \textbf{Rechnung:} \\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -\frac{8}{49}x^{2}-\frac{24}{49}x+1\frac{31}{49} =0 \\ x_{1/2}=\displaystyle\frac{+\frac{24}{49} \pm\sqrt{\left(-\frac{24}{49}\right)^{2}-4\cdot \left(-\frac{8}{49}\right) \cdot 1\frac{31}{49}}}{2\cdot\left(-\frac{8}{49}\right)} \\ x_{1/2}=\displaystyle \frac{+\frac{24}{49} \pm\sqrt{1\frac{15}{49}}}{-\frac{16}{49}} \\ x_{1/2}=\displaystyle \frac{\frac{24}{49} \pm1\frac{1}{7}}{-\frac{16}{49}} \\ x_{1}=\displaystyle \frac{\frac{24}{49} +1\frac{1}{7}}{-\frac{16}{49}} \qquad x_{2}=\displaystyle \frac{\frac{24}{49} -1\frac{1}{7}}{-\frac{16}{49}} \\ x_{1}=-5 \qquad x_{2}=2 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -\frac{8}{49}x^{2}-\frac{24}{49}x+1\frac{31}{49} =0 \qquad /:-\frac{8}{49} \\ x^{2}+3x-10 =0 \\ x_{1/2}=\displaystyle -\frac{3}{2}\pm\sqrt{\left(\frac{3}{2}\right)^2- \left(-10\right)} \\ x_{1/2}=\displaystyle -1\frac{1}{2}\pm\sqrt{12\frac{1}{4}} \\ x_{1/2}=\displaystyle -1\frac{1}{2}\pm3\frac{1}{2} \\ x_{1}=2 \qquad x_{2}=-5 \end{array}\\ \end{array}$