38756928539284314123049008318026368438221998823818157701981611847981052946086230517779418600353541323118851294249609652880788683611489679696027792584816798286546439597625664031512417580575356281435570813989103921311479584760379287707724243274506906164504370947981399049559728910046659604440048209472 104536113776305096559688240973949256190167194032892185858412761355595972804612264795071065166934350812256523818708562538374700528878580738296832597039986338236668384742302941726521217771506223372597117591535962383854860697754550083442105428640948558383002934553977298669928578507173938623814610729897 {\displaystyle 38756928539284314123049008318026368438221998823818157701981611847981052946086230517779418600353541323118851294249609652880788683611489679696027792584816798286546439597625664031512417580575356281435570813989103921311479584760379287707724243274506906164504370947981399049559728910046659604440048209472^{104536113776305096559688240973949256190167194032892185858412761355595972804612264795071065166934350812256523818708562538374700528878580738296832597039986338236668384742302941726521217771506223372597117591535962383854860697754550083442105428640948558383002934553977298669928578507173938623814610729897}\,}
( 38756928539284314123049008318026368438221998823818157701981611847981052946086230517779418600353541323118851294249609652880788683611489679696027792584816798286546439597625664031512417580575356281435570813989103921311479584760379287707724243274506906164504370947981399049559728910046659604440048209472 104536113776305096559688240973949256190167194032892185858412761355595972804612264795071065166934350812256523818708562538374700528878580738296832597039986338236668384742302941726521217771506223372597117591535962383854860697754550083442105428640948558383002934553977298669928578507173938623814610729897 105966046330769304886073092822589624299830328759415167682442574582167794055871049254769493025329667962976212114266006479387868344184327682711205269031458046574665080524048024146643248425659407860467229982524540867201511530051391067636012941954138057614734658851306048235208397343325303969187863015429 ) {\displaystyle \left({\frac {38756928539284314123049008318026368438221998823818157701981611847981052946086230517779418600353541323118851294249609652880788683611489679696027792584816798286546439597625664031512417580575356281435570813989103921311479584760379287707724243274506906164504370947981399049559728910046659604440048209472^{104536113776305096559688240973949256190167194032892185858412761355595972804612264795071065166934350812256523818708562538374700528878580738296832597039986338236668384742302941726521217771506223372597117591535962383854860697754550083442105428640948558383002934553977298669928578507173938623814610729897}}{105966046330769304886073092822589624299830328759415167682442574582167794055871049254769493025329667962976212114266006479387868344184327682711205269031458046574665080524048024146643248425659407860467229982524540867201511530051391067636012941954138057614734658851306048235208397343325303969187863015429}}\right)}
( y ( G G − G ) ( x ) − y ( e x − G y ( G ) ( x ) ) ( x ) ( y ( x ) − y ′ ( x ) e [ y ( x ) + y ″ ( x ) ] ) y ( G x ) ( x − 1 ) + 1 ( y ( x ) ( x ) ) ) − ( ∑ k = 1 ∞ y ( x ) ( x − k ) ) ( y ′ ( x ) − y ‴ ( x ) ) ( y ‴ ( x ) + e y ‴ ( x ) ) + 1 = 0 {\displaystyle \left({\sqrt[{\left(y^{(x)}(x)\right)}]{\cfrac {y^{(G^{G}-G)}(x)-y^{(e^{x}-Gy^{(G)}(x))}(x)}{\left({\cfrac {y(x)-y'(x)}{e^{[y(x)+y''(x)]}}}\right)y^{(Gx)}(x-1)+1}}}\right)-\left(\sum _{k=1}^{\infty }y^{(x)}(x-k)\right)^{(y'(x)-y'''(x))^{(y'''(x)+e^{y'''(x)})}}+1=0}
y ( x ) ( x ) = e x {\displaystyle \ y^{(x)}(x)=e^{x}}
y ( x ) = e x {\displaystyle \ y(x)=e^{x}}
⨌ ⨌ ⨌ ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t d x d y d z d t ⨌ ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t d x d y d z d t d x d y d z d t ⨌ ⨌ ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t d x d y d z d t ⨌ ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t ⨌ ⨌ F U d x d y d z d t ⨌ F U d x d y d z d t d x d y d z d t d x d y d z d t d x d y d z d t d x d y d z d t {\displaystyle \iiiint _{\iiiint _{\iiiint _{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}^{\iiiint _{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}^{\iiiint _{F}^{U}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}\,dx\,dy\,dz\,dt}
a x + b = 0 , r 1 = − b a a x 2 + b x + c = 0 , r 1 = − b + b 2 − 4 a c 2 a , r 2 = − b − b 2 − 4 a c 2 a , a x 3 + b x 2 + c x + d = 0 , r 1 = − b 3 a − 1 3 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d + ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 − 1 3 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d − ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 r 2 = − b 3 a + 1 + i 3 6 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d + ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 + 1 − i 3 6 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d − ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 r 3 = − b 3 a + 1 − i 3 6 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d + ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 + 1 + i 3 6 a 1 2 [ 2 b 3 − 9 a b c + 27 a 2 d − ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 − 4 ( b 2 − 3 a c ) 3 ] 3 x 4 + a x 3 + b x 2 + c x + d = 0 , 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}&ax+b=0,\\&r_{1}={\frac {-b}{a}}\\&ax^{2}+bx+c=0,\\&r_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\\&r_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},\\&ax^{3}+bx^{2}+cx+d=0,\\&r_{1}=-{\frac {b}{3a}}-{\frac {1}{3a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}-{\frac {1}{3a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&r_{2}=-{\frac {b}{3a}}+{\frac {1+i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}+{\frac {1-i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&r_{3}=-{\frac {b}{3a}}+{\frac {1-i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}+{\frac {1+i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&x^{4}+ax^{3}+bx^{2}+cx+d=0,\\&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}}}
![{\displaystyle \left({\sqrt[{\left(y^{(x)}(x)\right)}]{\cfrac {y^{(G^{G}-G)}(x)-y^{(e^{x}-Gy^{(G)}(x))}(x)}{\left({\cfrac {y(x)-y'(x)}{e^{[y(x)+y''(x)]}}}\right)y^{(Gx)}(x-1)+1}}}\right)-\left(\sum _{k=1}^{\infty }y^{(x)}(x-k)\right)^{(y'(x)-y'''(x))^{(y'''(x)+e^{y'''(x)})}}+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b748b48640307daf7151b96097d7e6ffcf2bf457)
![{\displaystyle {\begin{aligned}&ax+b=0,\\&r_{1}={\frac {-b}{a}}\\&ax^{2}+bx+c=0,\\&r_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\\&r_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},\\&ax^{3}+bx^{2}+cx+d=0,\\&r_{1}=-{\frac {b}{3a}}-{\frac {1}{3a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}-{\frac {1}{3a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&r_{2}=-{\frac {b}{3a}}+{\frac {1+i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}+{\frac {1-i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&r_{3}=-{\frac {b}{3a}}+{\frac {1-i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d+{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}+{\frac {1+i{\sqrt {3}}}{6a}}{\sqrt[{3}]{{\tfrac {1}{2}}\left[2b^{3}-9abc+27a^{2}d-{\sqrt {\left(2b^{3}-9abc+27a^{2}d\right)^{2}-4\left(b^{2}-3ac\right)^{3}}}\right]}}\\&x^{4}+ax^{3}+bx^{2}+cx+d=0,\\&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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e242347afb8587f07704ee4ea8b6ceb8f06f82e1)
