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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

Ha introducido [src]
 3              
x  + 3*x + 2 = 0
$$\left(x^{3} + 3 x\right) + 2 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = 2$$
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
Suma y producto de raíces [src]
suma
                            _______________     /         _______________                       \                               _______________     /                                  _______________\                           _______________
                         3 /           ___      |  ___ 3 /           ___              ___       |                            3 /           ___      |            ___            ___ 3 /           ___ |                        3 /           ___ 
           3             \/  27 + 27*\/ 2       |\/ 3 *\/  27 + 27*\/ 2           3*\/ 3        |              3             \/  27 + 27*\/ 2       |        3*\/ 3           \/ 3 *\/  27 + 27*\/ 2  |           3            \/  27 + 27*\/ 2  
- -------------------- + ------------------ + I*|------------------------ + --------------------| + - -------------------- + ------------------ + I*|- -------------------- - ------------------------| + ------------------ - ------------------
       _______________           6              |           6                    _______________|          _______________           6              |       _______________              6            |      _______________           3         
    3 /           ___                           |                             3 /           ___ |       3 /           ___                           |    3 /           ___                            |   3 /           ___                      
  2*\/  27 + 27*\/ 2                            \                           2*\/  27 + 27*\/ 2  /     2*\/  27 + 27*\/ 2                            \  2*\/  27 + 27*\/ 2                             /   \/  27 + 27*\/ 2                       
$$\left(- \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}\right) + \left(\left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)\right) + \left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)\right)\right)$$
=
  /                                  _______________\     /         _______________                       \
  |            ___            ___ 3 /           ___ |     |  ___ 3 /           ___              ___       |
  |        3*\/ 3           \/ 3 *\/  27 + 27*\/ 2  |     |\/ 3 *\/  27 + 27*\/ 2           3*\/ 3        |
I*|- -------------------- - ------------------------| + I*|------------------------ + --------------------|
  |       _______________              6            |     |           6                    _______________|
  |    3 /           ___                            |     |                             3 /           ___ |
  \  2*\/  27 + 27*\/ 2                             /     \                           2*\/  27 + 27*\/ 2  /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right) + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)$$
producto
/                            _______________     /         _______________                       \\ /                            _______________     /                                  _______________\\ /                        _______________\
|                         3 /           ___      |  ___ 3 /           ___              ___       || |                         3 /           ___      |            ___            ___ 3 /           ___ || |                     3 /           ___ |
|           3             \/  27 + 27*\/ 2       |\/ 3 *\/  27 + 27*\/ 2           3*\/ 3        || |           3             \/  27 + 27*\/ 2       |        3*\/ 3           \/ 3 *\/  27 + 27*\/ 2  || |        3            \/  27 + 27*\/ 2  |
|- -------------------- + ------------------ + I*|------------------------ + --------------------||*|- -------------------- + ------------------ + I*|- -------------------- - ------------------------||*|------------------ - ------------------|
|       _______________           6              |           6                    _______________|| |       _______________           6              |       _______________              6            || |   _______________           3         |
|    3 /           ___                           |                             3 /           ___ || |    3 /           ___                           |    3 /           ___                            || |3 /           ___                      |
\  2*\/  27 + 27*\/ 2                            \                           2*\/  27 + 27*\/ 2  // \  2*\/  27 + 27*\/ 2                            \  2*\/  27 + 27*\/ 2                             // \\/  27 + 27*\/ 2                       /
$$\left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)\right) \left(- \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)\right) \left(- \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}\right)$$
=
-2
$$-2$$
-2
Respuesta rápida [src]
                                 _______________     /         _______________                       \
                              3 /           ___      |  ___ 3 /           ___              ___       |
                3             \/  27 + 27*\/ 2       |\/ 3 *\/  27 + 27*\/ 2           3*\/ 3        |
x1 = - -------------------- + ------------------ + I*|------------------------ + --------------------|
            _______________           6              |           6                    _______________|
         3 /           ___                           |                             3 /           ___ |
       2*\/  27 + 27*\/ 2                            \                           2*\/  27 + 27*\/ 2  /
$$x_{1} = - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right)$$
                                 _______________     /                                  _______________\
                              3 /           ___      |            ___            ___ 3 /           ___ |
                3             \/  27 + 27*\/ 2       |        3*\/ 3           \/ 3 *\/  27 + 27*\/ 2  |
x2 = - -------------------- + ------------------ + I*|- -------------------- - ------------------------|
            _______________           6              |       _______________              6            |
         3 /           ___                           |    3 /           ___                            |
       2*\/  27 + 27*\/ 2                            \  2*\/  27 + 27*\/ 2                             /
$$x_{2} = - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right)$$
                             _______________
                          3 /           ___ 
             3            \/  27 + 27*\/ 2  
x3 = ------------------ - ------------------
        _______________           3         
     3 /           ___                      
     \/  27 + 27*\/ 2                       
$$x_{3} = - \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} + \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}}$$
x3 = -(27 + 27*sqrt(2))^(1/3)/3 + 3/(27 + 27*sqrt(2))^(1/3)
Respuesta numérica [src]
x1 = -0.596071637983322
x2 = 0.298035818991661 + 1.80733949445202*i
x3 = 0.298035818991661 - 1.80733949445202*i
x3 = 0.298035818991661 - 1.80733949445202*i
Gráfico
x^3+3*x+2=0 la ecuación