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r 1 = − a 4 − 1 2 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − 1 2 a 2 2 − 4 b 3 − 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 − ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − − a 3 + 4 a b − 8 c 4 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 r 2 = − a 4 − 1 2 a 2 4 + 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − 1 2 a 2 2 − 4 b 3 − 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 − ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − − a 3 + 4 a b − 8 c 4 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 r 3 = − a 4 + 1 2 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − 1 2 a 2 2 − 4 b 3 − 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 − ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − − a 3 + 4 a b − 8 c 4 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 r 4 = − a 4 + 1 2 a 2 4 + 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − 1 2 a 2 2 − 4 b 3 − 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 − ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 − − a 3 + 4 a b − 8 c 4 a 2 4 − 2 b 3 + 2 1 3 ( b 2 − 3 a c + 12 d ) 3 ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 ) 1 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d + − 4 ( b 2 − 3 a c + 12 d ) 3 + ( 2 b 3 − 9 a b c + 27 c 2 + 27 a 2 d − 72 b d ) 2 54 ) 1 3 {\displaystyle {\begin{aligned}r_{1}&={\frac {-a}{4}}-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{2}&={\frac {-a}{4}}-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}+{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{3}&={\frac {-a}{4}}+{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\\r_{4}&={\frac {-a}{4}}+{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{4}}+{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}\\&-{\frac {1}{2}}{\sqrt {{\frac {a^{2}}{2}}-{\frac {4b}{3}}-{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}-\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}-{\frac {-a^{3}+4ab-8c}{4{\sqrt {{\frac {a^{2}}{4}}-{\frac {2b}{3}}+{\frac {2^{\frac {1}{3}}\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}\right)}^{\frac {1}{3}}}}+\left({\frac {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt {-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2}}}}{54}}\right)^{\frac {1}{3}}}}}}}}\end{aligned}}} <math>\begin{align} r_1 & =\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\ & -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\ r_2 & =\frac{-a}{4}-\frac{1}{2}{\sqrt{\frac{a^{2} }{4}+\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\ & -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\ r_3 & =\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\ & -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } } \\ r_4 & =\frac{-a}{4}+\frac{1}{2}{\sqrt{\frac{a^{2} }{4}+\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } \\ & -\frac{1}{2}{\sqrt{\frac{a^{2} }{2}-\frac{4b}{3}-\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }-\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} }-\frac{-a^{3}+4ab-8c}{4{\sqrt{\frac{a^{2} }{4}-\frac{2b}{3}+\frac{2^{\frac{1}{3} }\left(b^{2}-3ac+12d\right)}{3{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } }\right)}^{\frac{1}{3} } }+\left(\frac{ {2b^{3}-9abc+27c^{2}+27a^{2}d-72bd+{\sqrt{-4{\left(b^{2}-3ac+12d\right)}^{3}+{\left(2b^{3}-9abc+27c^{2}+27a^{2}d-72bd\right)}^{2} } } } }{54}\right)^{\frac{1}{3} } } } } } }
\end{align}</math>Deutsch ∙ English ∙ italiano ∙ magyar ∙ Nederlands ∙ polski ∙ sicilianu ∙ svenska ∙ Ελληνικά ∙ български ∙ македонски ∙ русский ∙ 日本語 ∙ فارسی ∙ +/−
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Lizenz Public domain Public domain false false
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Bengalisch All solutions of the equation x^4+ax^3+bx^2+cx+d=0
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Version vom Vorschaubild Maße Benutzer Kommentar aktuell 01:10, 17. Mai 2013 14.406 × 1.443 (326 KB) Linket {{subst:Upload marker added by en.wp UW}} {{Information |Description = {{en|All 4 roots of a quartic equation (x^4+ax^3+bx^2+cx+d=0).}} |Source = http://planetmath.org/quarticformula |Author = David Jao }} Category:Mathematical equations
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