Beispiel Nr: 07
$ \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^2-4 \qquad p_2: y=-1x^2+5 \\ \\ \textbf{Rechnung:} \\f\left(x\right)= x^2-4\qquad g\left(x\right)=-1x^2+5\\ \bullet \text{Schnittpunkte zwischen zwei Funktionen} \\ f\left(x\right)=g\left(x\right) \\ x^2-4=-1x^2+5 \\ x^2-4-(-1x^2+5)=0\\ \begin{array}{l|l|l|l} \begin{array}{l} \text{Umformen}\\ \hline 2x^2-9 =0 \qquad /+9 \\ 2x^2= 9 \qquad /:2 \\ x^2=\displaystyle\frac{9}{2} \\ x=\pm\sqrt{4\frac{1}{2}} \\ x_1=2,12 \qquad x_2=-2,12 \end{array}& \begin{array}{l} \text{a-b-c Formel}\\ \hline \\ 2x^{2}+0x-9 =0 \\ x_{1/2}=\displaystyle\frac{-0 \pm\sqrt{0^{2}-4\cdot 2 \cdot \left(-9\right)}}{2\cdot2} \\ x_{1/2}=\displaystyle \frac{-0 \pm\sqrt{72}}{4} \\ x_{1/2}=\displaystyle \frac{0 \pm8,49}{4} \\ x_{1}=\displaystyle \frac{0 +8,49}{4} \qquad x_{2}=\displaystyle \frac{0 -8,49}{4} \\ x_{1}=2,12 \qquad x_{2}=-2,12 \end{array}& \begin{array}{l} \text{p-q Formel}\\ \hline \\ 2x^{2}+0x-9 =0 \qquad /:2 \\ x^{2}+0x-4\frac{1}{2} =0 \\ x_{1/2}=\displaystyle -\frac{0}{2}\pm\sqrt{\left(\frac{0}{2}\right)^2- \left(-4\frac{1}{2}\right)} \\ x_{1/2}=\displaystyle 0\pm\sqrt{4\frac{1}{2}} \\ x_{1/2}=\displaystyle 0\pm2,12 \\ x_{1}=2,12 \qquad x_{2}=-2,12 \end{array}\\ \end{array}\\ \\ \text{Schnittpunkt }1\\ f(-2,12)=\frac{1}{2}\\ S(-2,12/\frac{1}{2})\\\\ \text{Schnittpunkt }2\\ f(2,12)=\frac{1}{2}\\ S(2,12/\frac{1}{2})\\$