Algebra-Gleichungen-Quadratische Gleichung
$ ax^{2}+bx+c=0 $
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Beispiel Nr: 45
$\begin{array}{l} \text{Gegeben:} ax^{2}+bx+c=0
\\ \text{Gesucht:} \\ \text{Lösung der Gleichung} \\
\\ ax^{2}+bx+c=0 \\ \textbf{Gegeben:} \\
\frac{8}{27}x^2+2\frac{2}{3}x =0
\\ \\ \textbf{Rechnung:} \\
\begin{array}{l|l|l|l}
\begin{array}{l}
\text{x-Ausklammern}\\ \hline
\frac{8}{27}x^{2}+2\frac{2}{3}x =0 \\
x(\frac{8}{27}x +2\frac{2}{3})=0 \\
\\ \frac{8}{27} x+2\frac{2}{3} =0 \qquad /-2\frac{2}{3} \\
\frac{8}{27} x= -2\frac{2}{3} \qquad /:\frac{8}{27} \\
x=\displaystyle\frac{-2\frac{2}{3}}{\frac{8}{27}}\\
x_1=0\\
x_2=-9
\end{array}&
\begin{array}{l}
\text{a-b-c Formel}\\ \hline
\\
\frac{8}{27}x^{2}+2\frac{2}{3}x+0 =0
\\
x_{1/2}=\displaystyle\frac{-2\frac{2}{3} \pm\sqrt{\left(2\frac{2}{3}\right)^{2}-4\cdot \frac{8}{27} \cdot 0}}{2\cdot\frac{8}{27}}
\\
x_{1/2}=\displaystyle \frac{-2\frac{2}{3} \pm\sqrt{7\frac{1}{9}}}{\frac{16}{27}}
\\
x_{1/2}=\displaystyle \frac{-2\frac{2}{3} \pm2\frac{2}{3}}{\frac{16}{27}}
\\
x_{1}=\displaystyle \frac{-2\frac{2}{3} +2\frac{2}{3}}{\frac{16}{27}} \qquad x_{2}=\displaystyle \frac{-2\frac{2}{3} -2\frac{2}{3}}{\frac{16}{27}}
\\
x_{1}=0 \qquad x_{2}=-9
\end{array}&
\begin{array}{l}
\text{p-q Formel}\\ \hline
\\
\frac{8}{27}x^{2}+2\frac{2}{3}x+0 =0 \qquad /:\frac{8}{27}
\\
x^{2}+9x+0 =0
\\
x_{1/2}=\displaystyle -\frac{9}{2}\pm\sqrt{\left(\frac{9}{2}\right)^2- 0}
\\
x_{1/2}=\displaystyle -4\frac{1}{2}\pm\sqrt{20\frac{1}{4}}
\\
x_{1/2}=\displaystyle -4\frac{1}{2}\pm4\frac{1}{2}
\\
x_{1}=0 \qquad x_{2}=-9
\end{array}\\ \end{array} \end{array}$