Title: Lösung[en] für klassische kubische Gleichungen
Course: Latex 004 - Beispiele

$$\displaystyle \left[x = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)}{\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3} - b^{2} c^{2} - 2 \,{\left(9 \, a b c - 2 \, b^{3}\right)} d} \sqrt{3}}{18 \, a^{2}} +\frac{-27 \, a^{2} d - 9 \, a b c + 2 \, b^{3}}{54 \,a^{3}}\right)}^{\left(\frac{1}{3}\right)} + \frac{-b}{3 \, a} +\frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(3 \, a c -b^{2}\right)}}{18 \, {\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3}- b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2 \, b^{3}\right)} d}\sqrt{3}}{18 \, a^{2}} + \frac{-27 \, a^{2} d - 9 \, a b c + 2 \,b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} a^{2}},\\x =-\frac{1}{2} \, {\left(-i \, \sqrt{3} + 1\right)} {\left(\frac{\sqrt{27\, a^{2} d^{2} + 4 \, a c^{3} - b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2\, b^{3}\right)} d} \sqrt{3}}{18 \, a^{2}} + \frac{-27 \, a^{2} d - 9 \,a b c + 2 \, b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} +\frac{-b}{3 \, a} + \frac{{\left(i \, \sqrt{3} + 1\right)} {\left(3 \, ac - b^{2}\right)}}{18 \, {\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, ac^{3} - b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2 \, b^{3}\right)} d}\sqrt{3}}{18 \, a^{2}} + \frac{-27 \, a^{2} d - 9 \, a b c + 2 \,b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} a^{2}}, \\x ={\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3} - b^{2} c^{2} - 2 \,{\left(9 \, a b c - 2 \, b^{3}\right)} d} \sqrt{3}}{18 \, a^{2}} +\frac{-27 \, a^{2} d - 9 \, a b c + 2 \, b^{3}}{54 \,a^{3}}\right)}^{\left(\frac{1}{3}\right)} + \frac{-b}{3 \, a} + \frac{-3\, a c - b^{2}}{9 \, {\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3}- b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2 \, b^{3}\right)} d}\sqrt{3}}{18 \, a^{2}} + \frac{-27 \, a^{2} d - 9 \, a b c + 2 \,b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} a^{2}}\right]$$