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x^3-3*x-1=0

x^3-3*x-1=0 la ecuación

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v

Solución numérica:

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Solución

Ha introducido [src] 
 3              
x  - 3*x - 1 = 0
(x3−3x)−1=0\left(x^{3} - 3 x\right) - 1 = 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=−1v = -1
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=−1x_{1} x_{2} x_{3} = -1
Gráfica
05-15-10-51015-50005000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
          /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                         
       cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                         
          \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \
x1 = - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------|
          2        |     2            2           |     _____________                ||         2           |     _____________                |
                   |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|
                   |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||
                   |                              |3 /   - + ------- *|- - - -------|||                     |3 /   - + ------- *|- - - -------||
                   \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   //
x1=−cos⁡(π9)2+re⁡(1(−12−3i2)12+3i23)+3sin⁡(π9)2+i(−3cos⁡(π9)2−sin⁡(π9)2+im⁡(1(−12−3i2)12+3i23))x_{1} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right) 
          /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                         
       cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                         
          \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \
x2 = - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------|
          2        |     2            2           |     _____________                ||         2           |     _____________                |
                   |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|
                   |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||
                   |                              |3 /   - + ------- *|- - + -------|||                     |3 /   - + ------- *|- - + -------||
                   \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   //
x2=re⁡(1(−12+3i2)12+3i23)−cos⁡(π9)2−3sin⁡(π9)2+i(im⁡(1(−12+3i2)12+3i23)−sin⁡(π9)2+3cos⁡(π9)2)x_{2} = \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) 
                               _____________
                              /         ___ 
             1               /  1   I*\/ 3  
x3 = ------------------ + 3 /   - + ------- 
          _____________   \/    2      2    
         /         ___                      
        /  1   I*\/ 3                       
     3 /   - + -------                      
     \/    2      2
x3=112+3i23+12+3i23x_{3} = \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}} 
x3 = (1/2 + sqrt(3)*i/2)^(-1/3) + (1/2 + sqrt(3)*i/2)^(1/3)
Suma y producto de raíces [src] 
suma
     /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                                 /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                                                      _____________
  cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                              cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                                                     /         ___ 
     \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \        \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \           1               /  1   I*\/ 3  
- ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------| + - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------| + ------------------ + 3 /   - + ------- 
     2        |     2            2           |     _____________                ||         2           |     _____________                |        2        |     2            2           |     _____________                ||         2           |     _____________                |        _____________   \/    2      2    
              |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|                 |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|       /         ___                      
              |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||                 |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||      /  1   I*\/ 3                       
              |                              |3 /   - + ------- *|- - - -------|||                     |3 /   - + ------- *|- - - -------||                 |                              |3 /   - + ------- *|- - + -------|||                     |3 /   - + ------- *|- - + -------||   3 /   - + -------                      
              \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   //                 \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   //   \/    2      2
((re⁡(1(−12+3i2)12+3i23)−cos⁡(π9)2−3sin⁡(π9)2+i(im⁡(1(−12+3i2)12+3i23)−sin⁡(π9)2+3cos⁡(π9)2))+(−cos⁡(π9)2+re⁡(1(−12−3i2)12+3i23)+3sin⁡(π9)2+i(−3cos⁡(π9)2−sin⁡(π9)2+im⁡(1(−12−3i2)12+3i23))))+(112+3i23+12+3i23)\left(\left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) + \left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(\frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right) 
=
                          _____________               /     /pi\     ___    /pi\                                         \     /     /pi\     ___    /pi\                                         \                                                                                  
                         /         ___                |  sin|--|   \/ 3 *cos|--|                                         |     |  sin|--|   \/ 3 *cos|--|                                         |                                                                                  
        1               /  1   I*\/ 3        /pi\     |     \9 /            \9 /     /                1                 \|     |     \9 /            \9 /     /                1                 \|     /                1                 \     /                1                 \
------------------ + 3 /   - + -------  - cos|--| + I*|- ------- + ------------- + im|----------------------------------|| + I*|- ------- - ------------- + im|----------------------------------|| + re|----------------------------------| + re|----------------------------------|
     _____________   \/    2      2          \9 /     |     2            2           |     _____________                ||     |     2            2           |     _____________                ||     |     _____________                |     |     _____________                |
    /         ___                                     |                              |    /         ___  /          ___\||     |                              |    /         ___  /          ___\||     |    /         ___  /          ___\|     |    /         ___  /          ___\|
   /  1   I*\/ 3                                      |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||     |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||     |   /  1   I*\/ 3   |  1   I*\/ 3 ||     |   /  1   I*\/ 3   |  1   I*\/ 3 ||
3 /   - + -------                                     |                              |3 /   - + ------- *|- - + -------|||     |                              |3 /   - + ------- *|- - - -------|||     |3 /   - + ------- *|- - + -------||     |3 /   - + ------- *|- - - -------||
\/    2      2                                        \                              \\/    2      2     \  2      2   ///     \                              \\/    2      2     \  2      2   ///     \\/    2      2     \  2      2   //     \\/    2      2     \  2      2   //
−cos⁡(π9)+re⁡(1(−12+3i2)12+3i23)+re⁡(1(−12−3i2)12+3i23)+112+3i23+i(im⁡(1(−12+3i2)12+3i23)−sin⁡(π9)2+3cos⁡(π9)2)+i(−3cos⁡(π9)2−sin⁡(π9)2+im⁡(1(−12−3i2)12+3i23))+12+3i23- \cos{\left(\frac{\pi}{9} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right) + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}} 
producto
/     /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                         \ /     /pi\     /     /pi\     ___    /pi\                                         \     ___    /pi\                                         \ /                          _____________\
|  cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                         | |  cos|--|     |  sin|--|   \/ 3 *cos|--|                                         |   \/ 3 *sin|--|                                         | |                         /         ___ |
|     \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \| |     \9 /     |     \9 /            \9 /     /                1                 \|            \9 /     /                1                 \| |        1               /  1   I*\/ 3  |
|- ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------||*|- ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------||*|------------------ + 3 /   - + ------- |
|     2        |     2            2           |     _____________                ||         2           |     _____________                || |     2        |     2            2           |     _____________                ||         2           |     _____________                || |     _____________   \/    2      2    |
|              |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|| |              |                              |    /         ___  /          ___\||                     |    /         ___  /          ___\|| |    /         ___                      |
|              |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||| |              |                              |   /  1   I*\/ 3   |  1   I*\/ 3 |||                     |   /  1   I*\/ 3   |  1   I*\/ 3 ||| |   /  1   I*\/ 3                       |
|              |                              |3 /   - + ------- *|- - - -------|||                     |3 /   - + ------- *|- - - -------||| |              |                              |3 /   - + ------- *|- - + -------|||                     |3 /   - + ------- *|- - + -------||| |3 /   - + -------                      |
\              \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   /// \              \                              \\/    2      2     \  2      2   ///                     \\/    2      2     \  2      2   /// \\/    2      2                         /
(−cos⁡(π9)2+re⁡(1(−12−3i2)12+3i23)+3sin⁡(π9)2+i(−3cos⁡(π9)2−sin⁡(π9)2+im⁡(1(−12−3i2)12+3i23)))(re⁡(1(−12+3i2)12+3i23)−cos⁡(π9)2−3sin⁡(π9)2+i(im⁡(1(−12+3i2)12+3i23)−sin⁡(π9)2+3cos⁡(π9)2))(112+3i23+12+3i23)\left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right) \left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(\frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right) 
=
  /       /     5*pi*I\  -5*pi*I\        
  |       |     ------|  -------|        
  |       |       9   |     18  |    /pi\
2*\-1 + I*\1 - e      /*e       /*cos|--|
                                     \9 /
2(−1+i(1−e5iπ9)e−5iπ18)cos⁡(π9)2 \left(-1 + i \left(1 - e^{\frac{5 i \pi}{9}}\right) e^{- \frac{5 i \pi}{18}}\right) \cos{\left(\frac{\pi}{9} \right)} 
2*(-1 + i*(1 - exp(5*pi*i/9))*exp(-5*pi*i/18))*cos(pi/9)
Respuesta numérica [src] 
x1 = 1.87938524157182
x2 = -1.53208888623796
x3 = -0.347296355333861
x3 = -0.347296355333861
Gráfico
x^3-3*x-1=0 la ecuación
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