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x^3+x+3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3            
x  + x + 3 = 0
(x3+x)+3=0\left(x^{3} + x\right) + 3 = 0
Simplificar
Teorema de Cardano-Vieta
es ecuación cúbica reducida
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=1q = 1
v=dav = \frac{d}{a}
v=3v = 3
Fórmulas de Cardano-Vieta
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=0x_{1} + x_{2} + x_{3} = 0
x1x2+x1x3+x2x3=1x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1
x1x2x3=3x_{1} x_{2} x_{3} = 3
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-20002000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
                                 ________________     /                                     ________________\                                    ________________     /                                       ________________\                                ________________
                                /          _____      |                                    /          _____ |                                   /          _____      |                                      /          _____ |                               /          _____ 
                               /  81   3*\/ 741       |                            ___    /  81   3*\/ 741  |                                  /  81   3*\/ 741       |                              ___    /  81   3*\/ 741  |                              /  81   3*\/ 741  
                            3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |                               3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |                           3 /   -- + --------- 
             1              \/    2        2          |         \/ 3                   \/    2        2     |                1              \/    2        2          |           \/ 3                   \/    2        2     |             1             \/    2        2     
- ----------------------- + --------------------- + I*|----------------------- + ---------------------------| + - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------| + --------------------- - ---------------------
         ________________             6               |       ________________                6             |            ________________             6               |         ________________                6             |        ________________             3          
        /          _____                              |      /          _____                               |           /          _____                              |        /          _____                               |       /          _____                         
       /  81   3*\/ 741                               |     /  81   3*\/ 741                                |          /  81   3*\/ 741                               |       /  81   3*\/ 741                                |      /  81   3*\/ 741                          
  2*3 /   -- + ---------                              |2*3 /   -- + ---------                               |     2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               |   3 /   -- + ---------                         
    \/    2        2                                  \  \/    2        2                                   /       \/    2        2                                  \    \/    2        2                                   /   \/    2        2
(−812+3741233+1812+374123)+((−12812+374123+812+3741236+i(−3812+3741236−32812+374123))+(−12812+374123+812+3741236+i(32812+374123+3812+3741236)))\left(- \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) + \left(\left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)\right) + \left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)\right)\right) 
=
  /                                     ________________\     /                                       ________________\
  |                                    /          _____ |     |                                      /          _____ |
  |                            ___    /  81   3*\/ 741  |     |                              ___    /  81   3*\/ 741  |
  |           ___            \/ 3 *3 /   -- + --------- |     |             ___            \/ 3 *3 /   -- + --------- |
  |         \/ 3                   \/    2        2     |     |           \/ 3                   \/    2        2     |
I*|----------------------- + ---------------------------| + I*|- ----------------------- - ---------------------------|
  |       ________________                6             |     |         ________________                6             |
  |      /          _____                               |     |        /          _____                               |
  |     /  81   3*\/ 741                                |     |       /  81   3*\/ 741                                |
  |2*3 /   -- + ---------                               |     |  2*3 /   -- + ---------                               |
  \  \/    2        2                                   /     \    \/    2        2                                   /
i(−3812+3741236−32812+374123)+i(32812+374123+3812+3741236)i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right) 
producto
/                                 ________________     /                                     ________________\\ /                                 ________________     /                                       ________________\\ /                             ________________\
|                                /          _____      |                                    /          _____ || |                                /          _____      |                                      /          _____ || |                            /          _____ |
|                               /  81   3*\/ 741       |                            ___    /  81   3*\/ 741  || |                               /  81   3*\/ 741       |                              ___    /  81   3*\/ 741  || |                           /  81   3*\/ 741  |
|                            3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- || |                            3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- || |                        3 /   -- + --------- |
|             1              \/    2        2          |         \/ 3                   \/    2        2     || |             1              \/    2        2          |           \/ 3                   \/    2        2     || |          1             \/    2        2     |
|- ----------------------- + --------------------- + I*|----------------------- + ---------------------------||*|- ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------||*|--------------------- - ---------------------|
|         ________________             6               |       ________________                6             || |         ________________             6               |         ________________                6             || |     ________________             3          |
|        /          _____                              |      /          _____                               || |        /          _____                              |        /          _____                               || |    /          _____                         |
|       /  81   3*\/ 741                               |     /  81   3*\/ 741                                || |       /  81   3*\/ 741                               |       /  81   3*\/ 741                                || |   /  81   3*\/ 741                          |
|  2*3 /   -- + ---------                              |2*3 /   -- + ---------                               || |  2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               || |3 /   -- + ---------                         |
\    \/    2        2                                  \  \/    2        2                                   // \    \/    2        2                                  \    \/    2        2                                   // \\/    2        2                             /
(−12812+374123+812+3741236+i(32812+374123+3812+3741236))(−12812+374123+812+3741236+i(−3812+3741236−32812+374123))(−812+3741233+1812+374123)\left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)\right) \left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) 
=
           __________________              ________________                           2/3                __________________                          2/3                                 2/3                     ________________                                     2/3
        3 /            _____        2/3 3 /          _____      3 ___ /         _____\          _____ 3 /            _____      3 ____ /       _____\      3 ___   _____ /         _____\         2/3   _____ 3 /          _____    3 ___ 6 ___   _____ /       _____\   
     81*\/  324 + 12*\/ 741     81*2   *\/  81 + 3*\/ 741     3*\/ 2 *\81 + 3*\/ 741 /      3*\/ 741 *\/  324 + 12*\/ 741     3*\/ 18 *\27 + \/ 741 /      \/ 2 *\/ 741 *\81 + 3*\/ 741 /      3*2   *\/ 741 *\/  81 + 3*\/ 741     \/ 2 *\/ 3 *\/ 247 *\27 + \/ 741 /   
-3 + ------------------------ - --------------------------- - --------------------------- - ------------------------------- + -------------------------- + --------------------------------- + ---------------------------------- - -------------------------------------
                32                           32                            8                               32                             8                                72                                  32                                     24
−3741324+12741332−81⋅22381+3741332−2324736(27+741)2324−323(81+3741)238−3+3183(27+741)238+23741(81+3741)2372+81324+12741332+3⋅22374181+3741332- \frac{3 \sqrt{741} \sqrt[3]{324 + 12 \sqrt{741}}}{32} - \frac{81 \cdot 2^{\frac{2}{3}} \sqrt[3]{81 + 3 \sqrt{741}}}{32} - \frac{\sqrt[3]{2} \sqrt{247} \sqrt[6]{3} \left(27 + \sqrt{741}\right)^{\frac{2}{3}}}{24} - \frac{3 \sqrt[3]{2} \left(81 + 3 \sqrt{741}\right)^{\frac{2}{3}}}{8} - 3 + \frac{3 \sqrt[3]{18} \left(27 + \sqrt{741}\right)^{\frac{2}{3}}}{8} + \frac{\sqrt[3]{2} \sqrt{741} \left(81 + 3 \sqrt{741}\right)^{\frac{2}{3}}}{72} + \frac{81 \sqrt[3]{324 + 12 \sqrt{741}}}{32} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{741} \sqrt[3]{81 + 3 \sqrt{741}}}{32} 
-3 + 81*(324 + 12*sqrt(741))^(1/3)/32 - 81*2^(2/3)*(81 + 3*sqrt(741))^(1/3)/32 - 3*2^(1/3)*(81 + 3*sqrt(741))^(2/3)/8 - 3*sqrt(741)*(324 + 12*sqrt(741))^(1/3)/32 + 3*18^(1/3)*(27 + sqrt(741))^(2/3)/8 + 2^(1/3)*sqrt(741)*(81 + 3*sqrt(741))^(2/3)/72 + 3*2^(2/3)*sqrt(741)*(81 + 3*sqrt(741))^(1/3)/32 - 2^(1/3)*3^(1/6)*sqrt(247)*(27 + sqrt(741))^(2/3)/24
Respuesta rápida [src] 
                                      ________________     /                                     ________________\
                                     /          _____      |                                    /          _____ |
                                    /  81   3*\/ 741       |                            ___    /  81   3*\/ 741  |
                                 3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |
                  1              \/    2        2          |         \/ 3                   \/    2        2     |
x1 = - ----------------------- + --------------------- + I*|----------------------- + ---------------------------|
              ________________             6               |       ________________                6             |
             /          _____                              |      /          _____                               |
            /  81   3*\/ 741                               |     /  81   3*\/ 741                                |
       2*3 /   -- + ---------                              |2*3 /   -- + ---------                               |
         \/    2        2                                  \  \/    2        2                                   /
x1=−12812+374123+812+3741236+i(32812+374123+3812+3741236)x_{1} = - \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right) 
                                      ________________     /                                       ________________\
                                     /          _____      |                                      /          _____ |
                                    /  81   3*\/ 741       |                              ___    /  81   3*\/ 741  |
                                 3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |
                  1              \/    2        2          |           \/ 3                   \/    2        2     |
x2 = - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
              ________________             6               |         ________________                6             |
             /          _____                              |        /          _____                               |
            /  81   3*\/ 741                               |       /  81   3*\/ 741                                |
       2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               |
         \/    2        2                                  \    \/    2        2                                   /
x2=−12812+374123+812+3741236+i(−3812+3741236−32812+374123)x_{2} = - \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) 
                                  ________________
                                 /          _____ 
                                /  81   3*\/ 741  
                             3 /   -- + --------- 
               1             \/    2        2     
x3 = --------------------- - ---------------------
          ________________             3          
         /          _____                         
        /  81   3*\/ 741                          
     3 /   -- + ---------                         
     \/    2        2
x3=−812+3741233+1812+374123x_{3} = - \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} 
x3 = -(81/2 + 3*sqrt(741)/2)^(1/3)/3 + (81/2 + 3*sqrt(741)/2)^(-1/3)
Respuesta numérica [src] 
x1 = 0.606705831381115 + 1.45061224918844*i
x2 = 0.606705831381115 - 1.45061224918844*i
x3 = -1.21341166276223
x3 = -1.21341166276223
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