Archivo Historial del archivo Uso de archivos Uso de archivos global Metadatos Esta imagen muestra algún tipo de fórmula que podría convertirse a TeX . Almacenar fórmulas como imágenes hace que sea más difícil cambiarlas. TeX también ayuda a asegurarse de que todos usen la misma fuente y tamaño. Se ha propuesto un reemplazo: 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 {alineado} 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 \ rig ht)} {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 {\ izquierda (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 {\ izquierda (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 {\ izquierda (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 + {\ sq rt {-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 {alineado}}} En su artículo, reemplace la imagen con:
\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}
български | Deutsch | Ελληνικά | Inglés | فارسی | magyar | italiano | македонски | Nederlands | polski | русский | sicilianu | svenska | +/−
Resumen Licencia Dominio público Dominio público falso falso
Este trabajo no es elegible para derechos de autor y, por lo tanto, es de dominio público porque consiste en su totalidad en información que es propiedad común y no contiene autoría original .
inglés Todas las soluciones de la ecuación x ^ 4 + ax ^ 3 + bx ^ 2 + cx + d = 0
Historial del archivo Haga clic en una fecha / hora para ver el archivo tal como apareció en ese momento.
Fecha y hora Miniatura Dimensiones Usuario Comentario Actual 00:10, 17 de mayo de 2013 14,406 × 1,443 (326 KB) Linket {{subst: Subir marcador agregado por en.wp UW}} {{Información | Descripción = {{en | Las 4 raíces de una ecuación cuártica (x ^ 4 + ax ^ 3 + bx ^ 2 + cx + d = 0) .}} | Fuente = http://planetmath.org/quarticformula | Autor = David Jao}} Categoría: Ecuaciones matemáticas
Uso de archivos Las siguientes páginas de la Wikipedia en inglés utilizan este archivo (no se enumeran las páginas de otros proyectos):
Uso de archivos global Los siguientes wikis utilizan este archivo:
Uso en bn.wikipedia.org Uso en bs.wikipedia.org Funkcija četvrtog stepena Uso en de.wikipedia.org Grados de Polynom vierten Uso en fi.wikipedia.org Neljännen polinomifunktio asteen Uso en sh.wikipedia.org Funkcija četvrtog stepena Este archivo contiene información adicional, probablemente agregada desde la cámara digital o el escáner utilizado para crearlo o digitalizarlo.
Si el archivo se ha modificado desde su estado original, es posible que algunos detalles no reflejen completamente el archivo modificado.