$\text{Parabel-Parabel}$
$\begin{array}{l} \text{Gegeben: } \\ p_1: y=a_1x^{2}+b_1x+c_1 \qquad p_2: y=a_2x^{2}+b_2x+c_2\\ \text{Gesucht:Schnittpunkte zwischen 2 Parabeln} \\ \text{Parabel-Parabel}\\ \textbf{Gegeben:} \\ p_1: y= x^2+1\frac{1}{2}x-\frac{1}{2} \qquad p_2: y=-1x^2-2\frac{1}{2}x+5\frac{1}{2} \\ \\ \textbf{Rechnung:} \\f\left(x\right)= x^2+1\frac{1}{2}x-\frac{1}{2}\qquad g\left(x\right)=-1x^2-2\frac{1}{2}x+5\frac{1}{2}\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\ x^2+1\frac{1}{2}x-\frac{1}{2}=-1x^2-2\frac{1}{2}x+5\frac{1}{2} \\ x^2+1\frac{1}{2}x-\frac{1}{2}-(-1x^2-2\frac{1}{2}x+5\frac{1}{2})=0\\ \begin{array}{l|l|l} \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 2x^{2}+4x-6 =0 \\ x_{1/2}=\displaystyle\frac{-4 \pm\sqrt{4^{2}-4\cdot 2 \cdot \left(-6\right)}}{2\cdot2} \\ x_{1/2}=\displaystyle \frac{-4 \pm\sqrt{64}}{4} \\ x_{1/2}=\displaystyle \frac{-4 \pm8}{4} \\ x_{1}=\displaystyle \frac{-4 +8}{4} \qquad x_{2}=\displaystyle \frac{-4 -8}{4} \\ x_{1}=1 \qquad x_{2}=-3 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ 2x^{2}+4x-6 =0 \qquad /:2 \\ x^{2}+2x-3 =0 \\ x_{1/2}=\displaystyle -\frac{2}{2}\pm\sqrt{\left(\frac{2}{2}\right)^2- \left(-3\right)} \\ x_{1/2}=\displaystyle -1\pm\sqrt{4} \\ x_{1/2}=\displaystyle -1\pm2 \\ x_{1}=1 \qquad x_{2}=-3 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-3)=4\\ S(-3/4)\\\\ \text{Schnittpunkt }2\\ f(1)=2\\ S(1/2)\\ \end{array}$