$\begin{array}{l} \text{Gegeben: } \\ P: y=a_1x^{2}+b_1x+c_1 \qquad g:y=mx+t\\ \text{Gesucht:Schnittpunkte zwischen Parabel und Gerade} \\ \text{Parabel-Gerade}\\ \textbf{Gegeben:} \\ p: y==-1x^2-5x \qquad g: y==-\frac{1}{2}x+2 \\ \\ \textbf{Rechnung:} \\f\left(x\right)=-1x^2-5x\qquad g\left(x\right)=-\frac{1}{2}x+2\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\-1x^2-5x=-\frac{1}{2}x+2 \\ -1x^2-5x-(-\frac{1}{2}x+2)=0\\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -1x^{2}-4\frac{1}{2}x-2 =0 \\ x_{1/2}=\displaystyle\frac{+4\frac{1}{2} \pm\sqrt{\left(-4\frac{1}{2}\right)^{2}-4\cdot \left(-1\right) \cdot \left(-2\right)}}{2\cdot\left(-1\right)} \\ x_{1/2}=\displaystyle \frac{+4\frac{1}{2} \pm\sqrt{12\frac{1}{4}}}{-2} \\ x_{1/2}=\displaystyle \frac{4\frac{1}{2} \pm3\frac{1}{2}}{-2} \\ x_{1}=\displaystyle \frac{4\frac{1}{2} +3\frac{1}{2}}{-2} \qquad x_{2}=\displaystyle \frac{4\frac{1}{2} -3\frac{1}{2}}{-2} \\ x_{1}=-4 \qquad x_{2}=-\frac{1}{2} \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -1x^{2}-4\frac{1}{2}x-2 =0 \qquad /:-1 \\ x^{2}+4\frac{1}{2}x+2 =0 \\ x_{1/2}=\displaystyle -\frac{4\frac{1}{2}}{2}\pm\sqrt{\left(\frac{4\frac{1}{2}}{2}\right)^2- 2} \\ x_{1/2}=\displaystyle -2\frac{1}{4}\pm\sqrt{3\frac{1}{16}} \\ x_{1/2}=\displaystyle -2\frac{1}{4}\pm1\frac{3}{4} \\ x_{1}=-\frac{1}{2} \qquad x_{2}=-4 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-4)=4\\ S(-4/4)\\\\ \text{Schnittpunkt }2\\ f(-\frac{1}{2})=2\frac{1}{4}\\ S(-\frac{1}{2}/2\frac{1}{4})\\ \end{array}$