$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y= 1\frac{1}{2}x^2-6x+3 \qquad p_2: y= \frac{1}{2}x^2-2x \\ \\ \textbf{Rechnung:} \\f\left(x\right)= 1\frac{1}{2}x^2-6x+3\qquad g\left(x\right)= \frac{1}{2}x^2-2x\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\ 1\frac{1}{2}x^2-6x+3= \frac{1}{2}x^2-2x \\ 1\frac{1}{2}x^2-6x+3-( \frac{1}{2}x^2-2x)=0\\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 1x^{2}-4x+3 =0 \\ x_{1/2}=\displaystyle\frac{+4 \pm\sqrt{\left(-4\right)^{2}-4\cdot 1 \cdot 3}}{2\cdot1} \\ x_{1/2}=\displaystyle \frac{+4 \pm\sqrt{4}}{2} \\ x_{1/2}=\displaystyle \frac{4 \pm2}{2} \\ x_{1}=\displaystyle \frac{4 +2}{2} \qquad x_{2}=\displaystyle \frac{4 -2}{2} \\ x_{1}=3 \qquad x_{2}=1 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ \\ x^{2}-4x+3 =0 \\ x_{1/2}=\displaystyle -\frac{-4}{2}\pm\sqrt{\left(\frac{\left(-4\right)}{2}\right)^2- 3} \\ x_{1/2}=\displaystyle 2\pm\sqrt{1} \\ x_{1/2}=\displaystyle 2\pm1 \\ x_{1}=3 \qquad x_{2}=1 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(1)=-1\frac{1}{2}\\ S(1/-1\frac{1}{2})\\\\ \text{Schnittpunkt }2\\ f(3)=-1\frac{1}{2}\\ S(3/-1\frac{1}{2})\\ \end{array}$