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x^3+3*x-2=0

x^3+3*x-2=0 la ecuación

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Solución numérica:

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Solución

Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = -2$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = -2$$
Gráfica
Respuesta rápida [src]
                           ___________     /                              ___________\
                        3 /       ___      |         ___           ___ 3 /       ___ |
            1           \/  1 + \/ 2       |       \/ 3          \/ 3 *\/  1 + \/ 2  |
x1 = ---------------- - -------------- + I*|- ---------------- - --------------------|
          ___________         2            |       ___________            2          |
       3 /       ___                       |    3 /       ___                        |
     2*\/  1 + \/ 2                        \  2*\/  1 + \/ 2                         /
$$x_{1} = - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)$$
                           ___________     /                            ___________\
                        3 /       ___      |       ___           ___ 3 /       ___ |
            1           \/  1 + \/ 2       |     \/ 3          \/ 3 *\/  1 + \/ 2  |
x2 = ---------------- - -------------- + I*|---------------- + --------------------|
          ___________         2            |     ___________            2          |
       3 /       ___                       |  3 /       ___                        |
     2*\/  1 + \/ 2                        \2*\/  1 + \/ 2                         /
$$x_{2} = - \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)$$
        ___________                 
     3 /       ___          1       
x3 = \/  1 + \/ 2   - --------------
                         ___________
                      3 /       ___ 
                      \/  1 + \/ 2  
$$x_{3} = - \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}$$
x3 = -1/(1 + sqrt(2))^(1/3) + (1 + sqrt(2))^(1/3)
Suma y producto de raíces [src]
suma
                      ___________     /                              ___________\                         ___________     /                            ___________\                                  
                   3 /       ___      |         ___           ___ 3 /       ___ |                      3 /       ___      |       ___           ___ 3 /       ___ |      ___________                 
       1           \/  1 + \/ 2       |       \/ 3          \/ 3 *\/  1 + \/ 2  |          1           \/  1 + \/ 2       |     \/ 3          \/ 3 *\/  1 + \/ 2  |   3 /       ___          1       
---------------- - -------------- + I*|- ---------------- - --------------------| + ---------------- - -------------- + I*|---------------- + --------------------| + \/  1 + \/ 2   - --------------
     ___________         2            |       ___________            2          |        ___________         2            |     ___________            2          |                       ___________
  3 /       ___                       |    3 /       ___                        |     3 /       ___                       |  3 /       ___                        |                    3 /       ___ 
2*\/  1 + \/ 2                        \  2*\/  1 + \/ 2                         /   2*\/  1 + \/ 2                        \2*\/  1 + \/ 2                         /                    \/  1 + \/ 2  
$$\left(- \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right) + \left(\left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)\right) + \left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)\right)\right)$$
=
  /                            ___________\     /                              ___________\
  |       ___           ___ 3 /       ___ |     |         ___           ___ 3 /       ___ |
  |     \/ 3          \/ 3 *\/  1 + \/ 2  |     |       \/ 3          \/ 3 *\/  1 + \/ 2  |
I*|---------------- + --------------------| + I*|- ---------------- - --------------------|
  |     ___________            2          |     |       ___________            2          |
  |  3 /       ___                        |     |    3 /       ___                        |
  \2*\/  1 + \/ 2                         /     \  2*\/  1 + \/ 2                         /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)$$
producto
/                      ___________     /                              ___________\\ /                      ___________     /                            ___________\\                                  
|                   3 /       ___      |         ___           ___ 3 /       ___ || |                   3 /       ___      |       ___           ___ 3 /       ___ || /   ___________                 \
|       1           \/  1 + \/ 2       |       \/ 3          \/ 3 *\/  1 + \/ 2  || |       1           \/  1 + \/ 2       |     \/ 3          \/ 3 *\/  1 + \/ 2  || |3 /       ___          1       |
|---------------- - -------------- + I*|- ---------------- - --------------------||*|---------------- - -------------- + I*|---------------- + --------------------||*|\/  1 + \/ 2   - --------------|
|     ___________         2            |       ___________            2          || |     ___________         2            |     ___________            2          || |                    ___________|
|  3 /       ___                       |    3 /       ___                        || |  3 /       ___                       |  3 /       ___                        || |                 3 /       ___ |
\2*\/  1 + \/ 2                        \  2*\/  1 + \/ 2                         // \2*\/  1 + \/ 2                        \2*\/  1 + \/ 2                         // \                 \/  1 + \/ 2  /
$$\left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{1 + \sqrt{2}}}{2} + \frac{1}{2 \sqrt[3]{1 + \sqrt{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{1 + \sqrt{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{1 + \sqrt{2}}}\right)\right) \left(- \frac{1}{\sqrt[3]{1 + \sqrt{2}}} + \sqrt[3]{1 + \sqrt{2}}\right)$$
=
2
$$2$$
2
Respuesta numérica [src]
x1 = -0.298035818991661 + 1.80733949445202*i
x2 = -0.298035818991661 - 1.80733949445202*i
x3 = 0.596071637983322
x3 = 0.596071637983322
Gráfico
x^3+3*x-2=0 la ecuación