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x^3-3*x+3=0

x^3-3*x+3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
 3              
x  - 3*x + 3 = 0
$$\left(x^{3} - 3 x\right) + 3 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = 3$$
Gráfica
Suma y producto de raíces [src]
suma
     _______________                              /                                      _______________\        _______________                              /             _______________                         \                                 _______________
    /           ___                               |                                     /           ___ |       /           ___                               |            /           ___                          |                                /           ___ 
   /  81   27*\/ 5                                |                             ___    /  81   27*\/ 5  |      /  81   27*\/ 5                                |    ___    /  81   27*\/ 5                           |                               /  81   27*\/ 5  
3 /   -- + --------                               |             ___           \/ 3 *3 /   -- + -------- |   3 /   -- + --------                               |  \/ 3 *3 /   -- + --------               ___        |                            3 /   -- + -------- 
\/    2       2                  3                |         3*\/ 3                  \/    2       2     |   \/    2       2                  3                |        \/    2       2               3*\/ 3         |              3             \/    2       2     
-------------------- + ---------------------- + I*|- ---------------------- + --------------------------| + -------------------- + ---------------------- + I*|- -------------------------- + ----------------------| + - -------------------- - --------------------
         6                    _______________     |         _______________               6             |            6                    _______________     |              6                       _______________|          _______________            3          
                             /           ___      |        /           ___                              |                                /           ___      |                                     /           ___ |         /           ___                        
                            /  81   27*\/ 5       |       /  81   27*\/ 5                               |                               /  81   27*\/ 5       |                                    /  81   27*\/ 5  |        /  81   27*\/ 5                         
                       2*3 /   -- + --------      |  2*3 /   -- + --------                              |                          2*3 /   -- + --------      |                               2*3 /   -- + -------- |     3 /   -- + --------                        
                         \/    2       2          \    \/    2       2                                  /                            \/    2       2          \                                 \/    2       2     /     \/    2       2                            
$$\left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right) + \left(\left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) + \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right)\right)$$
=
  /                                      _______________\     /             _______________                         \
  |                                     /           ___ |     |            /           ___                          |
  |                             ___    /  81   27*\/ 5  |     |    ___    /  81   27*\/ 5                           |
  |             ___           \/ 3 *3 /   -- + -------- |     |  \/ 3 *3 /   -- + --------               ___        |
  |         3*\/ 3                  \/    2       2     |     |        \/    2       2               3*\/ 3         |
I*|- ---------------------- + --------------------------| + I*|- -------------------------- + ----------------------|
  |         _______________               6             |     |              6                       _______________|
  |        /           ___                              |     |                                     /           ___ |
  |       /  81   27*\/ 5                               |     |                                    /  81   27*\/ 5  |
  |  2*3 /   -- + --------                              |     |                               2*3 /   -- + -------- |
  \    \/    2       2                                  /     \                                 \/    2       2     /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right) + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
producto
/     _______________                              /                                      _______________\\ /     _______________                              /             _______________                         \\ /                              _______________\
|    /           ___                               |                                     /           ___ || |    /           ___                               |            /           ___                          || |                             /           ___ |
|   /  81   27*\/ 5                                |                             ___    /  81   27*\/ 5  || |   /  81   27*\/ 5                                |    ___    /  81   27*\/ 5                           || |                            /  81   27*\/ 5  |
|3 /   -- + --------                               |             ___           \/ 3 *3 /   -- + -------- || |3 /   -- + --------                               |  \/ 3 *3 /   -- + --------               ___        || |                         3 /   -- + -------- |
|\/    2       2                  3                |         3*\/ 3                  \/    2       2     || |\/    2       2                  3                |        \/    2       2               3*\/ 3         || |           3             \/    2       2     |
|-------------------- + ---------------------- + I*|- ---------------------- + --------------------------||*|-------------------- + ---------------------- + I*|- -------------------------- + ----------------------||*|- -------------------- - --------------------|
|         6                    _______________     |         _______________               6             || |         6                    _______________     |              6                       _______________|| |       _______________            3          |
|                             /           ___      |        /           ___                              || |                             /           ___      |                                     /           ___ || |      /           ___                        |
|                            /  81   27*\/ 5       |       /  81   27*\/ 5                               || |                            /  81   27*\/ 5       |                                    /  81   27*\/ 5  || |     /  81   27*\/ 5                         |
|                       2*3 /   -- + --------      |  2*3 /   -- + --------                              || |                       2*3 /   -- + --------      |                               2*3 /   -- + -------- || |  3 /   -- + --------                        |
\                         \/    2       2          \    \/    2       2                                  // \                         \/    2       2          \                                 \/    2       2     // \  \/    2       2                            /
$$\left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right) \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)$$
=
-3
$$-3$$
-3
Respuesta rápida [src]
          _______________                              /                                      _______________\
         /           ___                               |                                     /           ___ |
        /  81   27*\/ 5                                |                             ___    /  81   27*\/ 5  |
     3 /   -- + --------                               |             ___           \/ 3 *3 /   -- + -------- |
     \/    2       2                  3                |         3*\/ 3                  \/    2       2     |
x1 = -------------------- + ---------------------- + I*|- ---------------------- + --------------------------|
              6                    _______________     |         _______________               6             |
                                  /           ___      |        /           ___                              |
                                 /  81   27*\/ 5       |       /  81   27*\/ 5                               |
                            2*3 /   -- + --------      |  2*3 /   -- + --------                              |
                              \/    2       2          \    \/    2       2                                  /
$$x_{1} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
          _______________                              /             _______________                         \
         /           ___                               |            /           ___                          |
        /  81   27*\/ 5                                |    ___    /  81   27*\/ 5                           |
     3 /   -- + --------                               |  \/ 3 *3 /   -- + --------               ___        |
     \/    2       2                  3                |        \/    2       2               3*\/ 3         |
x2 = -------------------- + ---------------------- + I*|- -------------------------- + ----------------------|
              6                    _______________     |              6                       _______________|
                                  /           ___      |                                     /           ___ |
                                 /  81   27*\/ 5       |                                    /  81   27*\/ 5  |
                            2*3 /   -- + --------      |                               2*3 /   -- + -------- |
                              \/    2       2          \                                 \/    2       2     /
$$x_{2} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)$$
                                   _______________
                                  /           ___ 
                                 /  81   27*\/ 5  
                              3 /   -- + -------- 
                3             \/    2       2     
x3 = - -------------------- - --------------------
            _______________            3          
           /           ___                        
          /  81   27*\/ 5                         
       3 /   -- + --------                        
       \/    2       2                            
$$x_{3} = - \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}$$
x3 = -(27*sqrt(5)/2 + 81/2)^(1/3)/3 - 3/(27*sqrt(5)/2 + 81/2)^(1/3)
Respuesta numérica [src]
x1 = -2.10380340273554
x2 = 1.05190170136777 + 0.565235851677171*i
x3 = 1.05190170136777 - 0.565235851677171*i
x3 = 1.05190170136777 - 0.565235851677171*i
Gráfico
x^3-3*x+3=0 la ecuación