$\text{Parabel-Parabel}$
$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y= 2x^2+4x \qquad p_2: y=-3x^2 \\ \\ \textbf{Rechnung:} \\f\left(x\right)= 2x^2+4x\qquad g\left(x\right)=-3x^2\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\ 2x^2+4x=-3x^2 \\ 2x^2+4x-(-3x^2)=0\\ \begin{array}{l|l|l|l} \begin{array}{l} \text{x-Ausklammern}\\ \hline 5x^{2}+4x =0 \\ x(5x +4)=0 \\ \\ 5 x+4 =0 \qquad /-4 \\ 5 x= -4 \qquad /:5 \\ x=\displaystyle\frac{-4}{5}\\ x_1=0\\ x_2=-\frac{4}{5} \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 5x^{2}+4x+0 =0 \\ x_{1/2}=\displaystyle\frac{-4 \pm\sqrt{4^{2}-4\cdot 5 \cdot 0}}{2\cdot5} \\ x_{1/2}=\displaystyle \frac{-4 \pm\sqrt{16}}{10} \\ x_{1/2}=\displaystyle \frac{-4 \pm4}{10} \\ x_{1}=\displaystyle \frac{-4 +4}{10} \qquad x_{2}=\displaystyle \frac{-4 -4}{10} \\ x_{1}=0 \qquad x_{2}=-\frac{4}{5} \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ 5x^{2}+4x+0 =0 \qquad /:5 \\ x^{2}+\frac{4}{5}x+0 =0 \\ x_{1/2}=\displaystyle -\frac{\frac{4}{5}}{2}\pm\sqrt{\left(\frac{\frac{4}{5}}{2}\right)^2- 0} \\ x_{1/2}=\displaystyle -\frac{2}{5}\pm\sqrt{\frac{4}{25}} \\ x_{1/2}=\displaystyle -\frac{2}{5}\pm\frac{2}{5} \\ x_{1}=0 \qquad x_{2}=-\frac{4}{5} \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-\frac{4}{5})=-1\frac{23}{25}\\ S(-\frac{4}{5}/-1\frac{23}{25})\\\\ \text{Schnittpunkt }2\\ f(0)=0\\ S(0/0)\\ \end{array}$