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x^3-3x-9=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3              
x  - 3*x - 9 = 0
(x3−3x)−9=0\left(x^{3} - 3 x\right) - 9 = 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=−3q = -3
v=dav = \frac{d}{a}
v=−9v = -9
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=−3x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3
x1x2x3=−9x_{1} x_{2} x_{3} = -9
Gráfica
-10.0-7.5-5.0-2.50.02.55.07.510.012.515.017.5-25002500
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
                             ____________     /                                 ____________\                                ____________     /           ____________                      \                                        
                            /       ____      |                                /       ____ |                               /       ____      |          /       ____                       |                                        
                           /  9   \/ 77       |                        ___    /  9   \/ 77  |                              /  9   \/ 77       |  ___    /  9   \/ 77                        |                            ____________
                        3 /   - + ------      |         ___          \/ 3 *3 /   - + ------ |                           3 /   - + ------      |\/ 3 *3 /   - + ------             ___       |                           /       ____ 
           1            \/    2     2         |       \/ 3                 \/    2     2    |              1            \/    2     2         |      \/    2     2              \/ 3        |           1              /  9   \/ 77  
- ------------------- - ----------------- + I*|------------------- - -----------------------| + - ------------------- - ----------------- + I*|----------------------- - -------------------| + ----------------- + 3 /   - + ------ 
         ____________           2             |       ____________              2           |            ____________           2             |           2                     ____________|        ____________   \/    2     2    
        /       ____                          |      /       ____                           |           /       ____                          |                                /       ____ |       /       ____                     
       /  9   \/ 77                           |     /  9   \/ 77                            |          /  9   \/ 77                           |                               /  9   \/ 77  |      /  9   \/ 77                      
  2*3 /   - + ------                          |2*3 /   - + ------                           |     2*3 /   - + ------                          |                          2*3 /   - + ------ |   3 /   - + ------                     
    \/    2     2                             \  \/    2     2                              /       \/    2     2                             \                            \/    2     2    /   \/    2     2
(1772+923+772+923)+((−772+9232−12772+923+i(−3772+9232+32772+923))+(−772+9232−12772+923+i(−32772+923+3772+9232)))\left(\frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)\right)\right) 
=
  /                                 ____________\     /           ____________                      \
  |                                /       ____ |     |          /       ____                       |
  |                        ___    /  9   \/ 77  |     |  ___    /  9   \/ 77                        |
  |         ___          \/ 3 *3 /   - + ------ |     |\/ 3 *3 /   - + ------             ___       |
  |       \/ 3                 \/    2     2    |     |      \/    2     2              \/ 3        |
I*|------------------- - -----------------------| + I*|----------------------- - -------------------|
  |       ____________              2           |     |           2                     ____________|
  |      /       ____                           |     |                                /       ____ |
  |     /  9   \/ 77                            |     |                               /  9   \/ 77  |
  |2*3 /   - + ------                           |     |                          2*3 /   - + ------ |
  \  \/    2     2                              /     \                            \/    2     2    /
i(−3772+9232+32772+923)+i(−32772+923+3772+9232)i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right) 
producto
/                             ____________     /                                 ____________\\ /                             ____________     /           ____________                      \\                                        
|                            /       ____      |                                /       ____ || |                            /       ____      |          /       ____                       ||                                        
|                           /  9   \/ 77       |                        ___    /  9   \/ 77  || |                           /  9   \/ 77       |  ___    /  9   \/ 77                        || /                         ____________\
|                        3 /   - + ------      |         ___          \/ 3 *3 /   - + ------ || |                        3 /   - + ------      |\/ 3 *3 /   - + ------             ___       || |                        /       ____ |
|           1            \/    2     2         |       \/ 3                 \/    2     2    || |           1            \/    2     2         |      \/    2     2              \/ 3        || |        1              /  9   \/ 77  |
|- ------------------- - ----------------- + I*|------------------- - -----------------------||*|- ------------------- - ----------------- + I*|----------------------- - -------------------||*|----------------- + 3 /   - + ------ |
|         ____________           2             |       ____________              2           || |         ____________           2             |           2                     ____________|| |     ____________   \/    2     2    |
|        /       ____                          |      /       ____                           || |        /       ____                          |                                /       ____ || |    /       ____                     |
|       /  9   \/ 77                           |     /  9   \/ 77                            || |       /  9   \/ 77                           |                               /  9   \/ 77  || |   /  9   \/ 77                      |
|  2*3 /   - + ------                          |2*3 /   - + ------                           || |  2*3 /   - + ------                          |                          2*3 /   - + ------ || |3 /   - + ------                     |
\    \/    2     2                             \  \/    2     2                              // \    \/    2     2                             \                            \/    2     2    // \\/    2     2                        /
(−772+9232−12772+923+i(−32772+923+3772+9232))(−772+9232−12772+923+i(−3772+9232+32772+923))(1772+923+772+923)\left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right)\right) \left(\frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right) 
=
9
99 
9
Respuesta rápida [src] 
                                  ____________     /                                 ____________\
                                 /       ____      |                                /       ____ |
                                /  9   \/ 77       |                        ___    /  9   \/ 77  |
                             3 /   - + ------      |         ___          \/ 3 *3 /   - + ------ |
                1            \/    2     2         |       \/ 3                 \/    2     2    |
x1 = - ------------------- - ----------------- + I*|------------------- - -----------------------|
              ____________           2             |       ____________              2           |
             /       ____                          |      /       ____                           |
            /  9   \/ 77                           |     /  9   \/ 77                            |
       2*3 /   - + ------                          |2*3 /   - + ------                           |
         \/    2     2                             \  \/    2     2                              /
x1=−772+9232−12772+923+i(−3772+9232+32772+923)x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right) 
                                  ____________     /           ____________                      \
                                 /       ____      |          /       ____                       |
                                /  9   \/ 77       |  ___    /  9   \/ 77                        |
                             3 /   - + ------      |\/ 3 *3 /   - + ------             ___       |
                1            \/    2     2         |      \/    2     2              \/ 3        |
x2 = - ------------------- - ----------------- + I*|----------------------- - -------------------|
              ____________           2             |           2                     ____________|
             /       ____                          |                                /       ____ |
            /  9   \/ 77                           |                               /  9   \/ 77  |
       2*3 /   - + ------                          |                          2*3 /   - + ------ |
         \/    2     2                             \                            \/    2     2    /
x2=−772+9232−12772+923+i(−32772+923+3772+9232)x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right) 
                              ____________
                             /       ____ 
             1              /  9   \/ 77  
x3 = ----------------- + 3 /   - + ------ 
          ____________   \/    2     2    
         /       ____                     
        /  9   \/ 77                      
     3 /   - + ------                     
     \/    2     2
x3=1772+923+772+923x_{3} = \frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}} 
x3 = (sqrt(77)/2 + 9/2)^(-1/3) + (sqrt(77)/2 + 9/2)^(1/3)
Respuesta numérica [src] 
x1 = 2.55414921860077
x2 = -1.27707460930039 - 1.37577566236622*i
x3 = -1.27707460930039 + 1.37577566236622*i
x3 = -1.27707460930039 + 1.37577566236622*i
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