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{\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>
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Public domain Public domain false false
English All solutions to the equation x^4+ax^3+bx^2+cx+d
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