Hallo,
erst einmal die Gleichung in die "Normalform" bringen:
\( x^{2}-\frac{u x}{u-v}+\frac{u^{2}}{u v-v^{2}}=\frac{u x}{v} \quad |-\frac{u x}{v} \)
\( x^{2}-\frac{u x}{u-v}-\frac{u x}{v}+\frac{u^{2}}{u v-v^{2}}=0 \)
\( x^{2}+\left(-\frac{u}{u-v}-\frac{u}{v}\right) \cdot x+\frac{u^{2}}{u v-v^{2}}=0 \)
\( x^{2}+\left(-\frac{u}{u-v}-\frac{u}{v}\right) \cdot x+\frac{u^{2}}{(u-v) \cdot v}=0 \)
\(x^2 -\frac{u^{2}}{(u-v) \cdot v} \cdot x+\frac{u^{2}}{(u-v) \cdot v}=0 \)
\( x^{2} \cdot(u-v) \cdot v-u^{2} x+u^{2}=0 \)
\( \left(u v-v^{2}\right) x^2-u^{2} x+u^{2}=0 \)
\(a=uv-v^2\quad b=-u^2\quad c=u^2\)
Mitternachtsformel:
\( x_{1 / 2}=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \)
\( =\frac{u^{2} \pm \sqrt{u^{4}-4\left(u v-v^{2}\right) \cdot u^{2}}}{2\left(u v-v^{2}\right)} \)
\( =\frac{u^{2} \pm \sqrt{u^{4}-4 u^{3} v+4 u^{2} v^{2}}}{2\left(u v-v^{2}\right)} \)
\( =\frac{u^{2} \pm \sqrt{\left(u^{2}-2 u v\right)^{2}}}{2\left(u v-v^{2}\right)} \)
\( =\frac{u^{2} \pm\left(u^{2}-2 u v\right)}{2\left(u v-v^{2}\right)} \)
1. Fall:
\( \frac{u^{2}+u^{2}-2 u v}{2\left(u v-v^{2}\right)}=\frac{2 u^{2}-2 u v}{2\left(u v-v^{2}\right)}=\frac{2 u(u-v)}{2\left(u v-v^{2}\right)} \)
\( =\frac{2 u(u-v)}{2 v(u-v)}=\frac{u}{v} \)
Den 2. Fall schaffst du sicherlich selber.
Gruß, Silvia
