我的沙盒：
testing
a x 2 + b x + c = 0 x 2 + b a x + c a = 0 x 2 + b a x + ( b 2 a ) 2 − ( b 2 a ) 2 + c a = 0 ( x + b 2 a ) 2 − b 2 4 a 2 + c a = 0 ( x + b 2 a ) 2 = b 2 4 a 2 − c a ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 2 a x = − b ± b 2 − 4 a c 2 a {\displaystyle {\begin{aligned}ax^{2}+bx+c&=0\\x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}-\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a^{2}}}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}-4ac}{4a^{2}}}\\x+{\frac {b}{2a}}&={\frac {\pm {\sqrt {b^{2}-4ac}}}{2a}}\\x&={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}}
x 1 = − b 4 a + 1 2 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 − 1 2 b 2 2 a 2 − 4 c 3 a − 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 − 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 3 a + − b 3 + 4 a b c − 8 a 2 d 4 a 3 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 {\displaystyle {x_{1}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}+{\frac {-b^{3}+4abc-8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}
x 2 = − b 4 a + 1 2 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 + 1 2 b 2 2 a 2 − 4 c 3 a − 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 − 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 − b 3 − 4 a b c + 8 a 2 d 4 a 3 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 {\displaystyle {x_{2}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}-{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}
x 3 = − b 4 a − 1 2 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 − 1 2 b 2 2 a 2 − 4 c 3 a − 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 − 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 + b 3 − 4 a b c + 8 a 2 d 4 a 3 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 {\displaystyle {x_{3}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}
x 4 = − b 4 a − 1 2 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 + 1 2 b 2 2 a 2 − 4 c 3 a − 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 − 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 + b 3 − 4 a b c + 8 a 2 d 4 a 3 ( b 2 a ) 2 − 2 c 3 a + 2 ( c 2 − 3 b d + 12 a e ) 3 3 a 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 + 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e + ( 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e ) 2 − 4 ( c 2 − 3 b d + 12 a e ) 3 3 3 2 a 3 {\displaystyle {x_{4}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}
x 8 = k {\displaystyle x^{8}=k} x i = k 1 / 8 ω i , i = 1 , … , 8 , {\displaystyle x_{i}=k^{1/8}\omega _{i}\,,\quad i=1,\dots ,8,}
![{\displaystyle {x_{1}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2}}a}}+{\frac {-b^{3}+4abc-8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0ecf3321b0eb565cef251044bf03bc9a42d5fd)
![{\displaystyle {x_{2}=-{\frac {b}{4a}}+{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}-{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5756a76a6de1d763cadaec8db415d0913ed21a)
![{\displaystyle {x_{3}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}-{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32778b8ad064e7b99a3669e4d964fd77c7ed44b8)
![{\displaystyle {x_{4}=-{\frac {b}{4a}}-{\frac {1}{2}}{\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {2c}{3a}}+{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}+{\frac {1}{2}}{\sqrt {{\frac {b^{2}}{2a^{2}}}-{\frac {4c}{3a}}-{\frac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}-{\frac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}+{\frac {b^{3}-4abc+8a^{2}d}{4a^{3}{\sqrt {\left({\dfrac {b}{2a}}\right)^{2}-{\dfrac {2c}{3a}}+{\dfrac {\sqrt[{3}]{2(c^{2}-3bd+12ae)}}{3a{\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}}}+{\dfrac {\sqrt[{3}]{2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace+{\sqrt {(2c^{3}-9bcd+27ad^{2}+27b^{2}e-72ace)^{2}-4(c^{2}-3bd+12ae)^{3}}}}}{3{\sqrt[{3}]{2a}}}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e4b43c5d09c7b0449851c129375acb02d5e509)
