$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y=-\frac{1}{2}x^2+4x+6 \qquad p_2: y=-2x^2-8x+2 \\ \\ \textbf{Rechnung:} \\f\left(x\right)=-\frac{1}{2}x^2+4x+6\qquad g\left(x\right)=-2x^2-8x+2\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\-\frac{1}{2}x^2+4x+6=-2x^2-8x+2 \\ -\frac{1}{2}x^2+4x+6-(-2x^2-8x+2)=0\\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 1\frac{1}{2}x^{2}+12x+4 =0 \\ x_{1/2}=\displaystyle\frac{-12 \pm\sqrt{12^{2}-4\cdot 1\frac{1}{2} \cdot 4}}{2\cdot1\frac{1}{2}} \\ x_{1/2}=\displaystyle \frac{-12 \pm\sqrt{120}}{3} \\ x_{1/2}=\displaystyle \frac{-12 \pm11}{3} \\ x_{1}=\displaystyle \frac{-12 +11}{3} \qquad x_{2}=\displaystyle \frac{-12 -11}{3} \\ x_{1}=-0,349 \qquad x_{2}=-7,65 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ 1\frac{1}{2}x^{2}+12x+4 =0 \qquad /:1\frac{1}{2} \\ x^{2}+8x+2\frac{2}{3} =0 \\ x_{1/2}=\displaystyle -\frac{8}{2}\pm\sqrt{\left(\frac{8}{2}\right)^2- 2\frac{2}{3}} \\ x_{1/2}=\displaystyle -4\pm\sqrt{13\frac{1}{3}} \\ x_{1/2}=\displaystyle -4\pm3,65 \\ x_{1}=-0,349 \qquad x_{2}=-7,65 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-7,65)=-53,9\\ S(-7,65/-53,9)\\\\ \text{Schnittpunkt }2\\ f(-0,349)=4,55\\ S(-0,349/4,55)\\ \end{array}$