\(\qquad\qquad\frac{8-x}2 -\frac{2x-11}{x-3}=\frac{x-2}6\)
\(\Longrightarrow\qquad\frac{(8-x)\cdot(x-3)}{2} -\frac{(2x-11)\cdot(x-3)}{x-3}=\frac{(x-2)\cdot(x-3)}6\)
\(\Longleftrightarrow\qquad\frac{-x^2+11x-24}{2} -2x+11=\frac{x^2-5x+6}6\)
\(\Longleftrightarrow\qquad-\frac12x^2+\frac{11}2x-12 -2x+11=\frac16x^2-\frac56x+1\)
\(\Longleftrightarrow\qquad-\frac12x^2+\frac72x-1=\frac16x^2-\frac56x+1\)
\(\Longleftrightarrow\qquad-\frac23x^2+4\frac13x-2=0\)
\(\Longleftrightarrow\qquad x^2-6\frac12x+3=0\)
Mit \(p\)-\(q\)-Formel:
\(x_{1/2}= \frac{13}4\pm\sqrt{\left(\frac{-13}4\right)^2-3}=\frac{13}4\pm \sqrt{\frac{169}{16}-\frac{48}{16}}=\frac{13}4\pm \sqrt{\frac{121}{16}}=\frac{13}4\pm\frac{11}{4}\)
Also \(x_1=\frac12\) und \(x_2=6\).
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