$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y= x^2+2x+2 \qquad p_2: y=-1x^2-2x+2 \\ \\ \textbf{Rechnung:} \\f\left(x\right)= x^2+2x+2\qquad g\left(x\right)=-1x^2-2x+2\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\ x^2+2x+2=-1x^2-2x+2 \\ x^2+2x+2-(-1x^2-2x+2)=0\\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline 2x^{2}+4x =0 \\ x(2x +4)=0 \\ \\ 2 x+4 =0 \qquad /-4 \\ 2 x= -4 \qquad /:2 \\ x=\displaystyle\frac{-4}{2}\\ x_1=0\\ x_2=-2 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 2x^{2}+4x+0 =0 \\ x_{1/2}=\displaystyle\frac{-4 \pm\sqrt{4^{2}-4\cdot 2 \cdot 0}}{2\cdot2} \\ x_{1/2}=\displaystyle \frac{-4 \pm\sqrt{16}}{4} \\ x_{1/2}=\displaystyle \frac{-4 \pm4}{4} \\ x_{1}=\displaystyle \frac{-4 +4}{4} \qquad x_{2}=\displaystyle \frac{-4 -4}{4} \\ x_{1}=0 \qquad x_{2}=-2 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ 2x^{2}+4x+0 =0 \qquad /:2 \\ x^{2}+2x+0 =0 \\ x_{1/2}=\displaystyle -\frac{2}{2}\pm\sqrt{\left(\frac{2}{2}\right)^2- 0} \\ x_{1/2}=\displaystyle -1\pm\sqrt{1} \\ x_{1/2}=\displaystyle -1\pm1 \\ x_{1}=0 \qquad x_{2}=-2 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-2)=2\\ S(-2/2)\\\\ \text{Schnittpunkt }2\\ f(0)=2\\ S(0/2)\\ \end{array}$