$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y=-1x^2+x+3 \qquad p_2: y= \frac{1}{2}x^2-4x+5 \\ \\ \textbf{Rechnung:} \\f\left(x\right)=-1x^2+x+3\qquad g\left(x\right)= \frac{1}{2}x^2-4x+5\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\-1x^2+x+3= \frac{1}{2}x^2-4x+5 \\ -1x^2+x+3-( \frac{1}{2}x^2-4x+5)=0\\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ -1\frac{1}{2}x^{2}+5x-2 =0 \\ x_{1/2}=\displaystyle\frac{-5 \pm\sqrt{5^{2}-4\cdot \left(-1\frac{1}{2}\right) \cdot \left(-2\right)}}{2\cdot\left(-1\frac{1}{2}\right)} \\ x_{1/2}=\displaystyle \frac{-5 \pm\sqrt{13}}{-3} \\ x_{1/2}=\displaystyle \frac{-5 \pm3,61}{-3} \\ x_{1}=\displaystyle \frac{-5 +3,61}{-3} \qquad x_{2}=\displaystyle \frac{-5 -3,61}{-3} \\ x_{1}=0,465 \qquad x_{2}=2,87 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ -1\frac{1}{2}x^{2}+5x-2 =0 \qquad /:-1\frac{1}{2} \\ x^{2}-3\frac{1}{3}x+1\frac{1}{3} =0 \\ x_{1/2}=\displaystyle -\frac{-3\frac{1}{3}}{2}\pm\sqrt{\left(\frac{\left(-3\frac{1}{3}\right)}{2}\right)^2- 1\frac{1}{3}} \\ x_{1/2}=\displaystyle 1\frac{2}{3}\pm\sqrt{1\frac{4}{9}} \\ x_{1/2}=\displaystyle 1\frac{2}{3}\pm1,2 \\ x_{1}=2,87 \qquad x_{2}=0,465 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(0,465)=3,25\\ S(0,465/3,25)\\\\ \text{Schnittpunkt }2\\ f(2,87)=-2,36\\ S(2,87/-2,36)\\ \end{array}$