$\begin{array}{l} \text{Gegeben: Formfaktor } a \text{und 2 Punkte } A(xa/ya)\qquad B(xb /yb) \\ \text{Gesucht:} \\ y=ax^{2}+bx+c \\ \\ \text{2 Punkte und Formfaktor}\\ \textbf{Gegeben:} \\ a=\frac{1}{2} \qquad A(\frac{1}{5}/2)\qquad B(\frac{2}{5}/5) \\\\ \\ \textbf{Rechnung:} \\ a=\frac{1}{2} \qquad A(\frac{1}{5}/2)\qquad B(\frac{2}{5}/5) \\ \text{Formfaktor a einsetzen:}\\ y=\frac{1}{2}x^{2}+bx+c \\ \begin{array}{ll|l} \text{I)Punkt A einsetzen}&& \text{II)Punkt B einsetzen}\\ 2=\frac{1}{2}\cdot\left(\frac{1}{5}\right)^{2}+b\cdot\frac{1}{5}+c & \qquad & 5=\frac{1}{2}\cdot\left(\frac{2}{5}\right)^{2}+b\cdot\frac{2}{5}+c \\ 2=\frac{1}{50}+\frac{1}{5} b+c \qquad /-\frac{1}{50} \qquad /-\frac{1}{5}b& \qquad & 5=\frac{2}{25}+\frac{2}{5} b+c \\ 2-\frac{1}{50}-\frac{1}{5} b=c &\qquad& 5=\frac{2}{25}+\frac{2}{5} b+c \\ 1\frac{49}{50}-\frac{1}{5} b=c &\qquad& 5=\frac{2}{25}+\frac{2}{5} b+c \\ \end{array}\\ \text{I in II}\\ \qquad 5=\frac{2}{25}+\frac{2}{5} b+ 1\frac{49}{50}-\frac{1}{5} b \\ \qquad 5=2\frac{3}{50}+\frac{1}{5} b \qquad /-2\frac{3}{50} \qquad /:\frac{1}{5} \\ b=\frac{5-2\frac{3}{50}}{\frac{1}{5}} \\ b=14\frac{7}{10} \\ c= 1\frac{49}{50}-\frac{1}{5} \cdot 14\frac{7}{10} \\ c=-\frac{24}{25} \\ y= \frac{1}{2}x^2+14\frac{7}{10}x-\frac{24}{25} \end{array}$