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3*x^3-12*x-8=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3               
3*x  - 12*x - 8 = 0
(3x3−12x)−8=0\left(3 x^{3} - 12 x\right) - 8 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(3x3−12x)−8=0\left(3 x^{3} - 12 x\right) - 8 = 0
de
ax3+bx2+cx+d=0a x^{3} + b x^{2} + c x + d = 0
como ecuación cúbica reducida
x3+bx2a+cxa+da=0x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0
x3−4x−83=0x^{3} - 4 x - \frac{8}{3} = 0
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=−4q = -4
v=dav = \frac{d}{a}
v=−83v = - \frac{8}{3}
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=−4x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4
x1x2x3=−83x_{1} x_{2} x_{3} = - \frac{8}{3}
Gráfica
05-15-10-51015-1000010000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
    /                1                 \     /                /                1                 \                \                                             /                1                 \     /    /                1                 \                          \                                                              
2*re|----------------------------------|     |            2*im|----------------------------------|                |                                         2*re|----------------------------------|     |2*im|----------------------------------|                          |                                                              
    |                     _____________|     |                |                     _____________|                |                                             |                     _____________|     |    |                     _____________|                          |                                                              
    |/          ___\     /         ___ |     |                |/          ___\     /         ___ |                |                                             |/          ___\     /         ___ |     |    |/          ___\     /         ___ |                          |                                                              
    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |                ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\|     ___    /pi\                             ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\          |     ___    /pi\          _____________                       
    ||- - - -------|*3 /   - + ------- |     |                ||- - - -------|*3 /   - + ------- |   \/ 3 *sin|--||   \/ 3 *cos|--|                             ||- - + -------|*3 /   - + ------- |     |    ||- - + -------|*3 /   - + ------- |   \/ 3 *sin|--|          |   \/ 3 *cos|--|         /         ___                        
    \\  2      2   / \/    6      18   /     |     /pi\       \\  2      2   / \/    6      18   /            \18/|            \18/      /pi\        /pi\       \\  2      2   / \/    6      18   /     |    \\  2      2   / \/    6      18   /            \18/      /pi\|            \18/        /  1   I*\/ 3              2          
---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--| + - sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - ------------- + 2*3 /   - + -------  + --------------------
                   3                         \     \18/                      3                             3      /         3            \18/        \18/                      3                         \                   3                             3            \18//         3           \/    6      18             _____________
                                                                                                                                                                                                                                                                                                                             /         ___ 
                                                                                                                                                                                                                                                                                                                            /  1   I*\/ 3  
                                                                                                                                                                                                                                                                                                                       3*3 /   - + ------- 
                                                                                                                                                                                                                                                                                                                         \/    6      18
((2re⁡(1(−12+3i2)16+3i183)3−3cos⁡(π18)3−sin⁡(π18)+i(2im⁡(1(−12+3i2)16+3i183)3−3sin⁡(π18)3+cos⁡(π18)))+(−3cos⁡(π18)3+2re⁡(1(−12−3i2)16+3i183)3+sin⁡(π18)+i(−cos⁡(π18)−3sin⁡(π18)3+2im⁡(1(−12−3i2)16+3i183)3)))+(2316+3i183+216+3i183)\left(\left(\frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right)\right) + \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right)\right)\right) + \left(\frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}\right) 
=
                                                  /                1                 \       /                1                 \     /                /                1                 \                \     /    /                1                 \                          \                  
                                              2*re|----------------------------------|   2*re|----------------------------------|     |            2*im|----------------------------------|                |     |2*im|----------------------------------|                          |                  
                                                  |                     _____________|       |                     _____________|     |                |                     _____________|                |     |    |                     _____________|                          |                  
                                                  |/          ___\     /         ___ |       |/          ___\     /         ___ |     |                |/          ___\     /         ___ |                |     |    |/          ___\     /         ___ |                          |                  
       _____________                              ||  1   I*\/ 3 |    /  1   I*\/ 3  |       ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |                ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\|     |    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\          |       ___    /pi\
      /         ___                               ||- - + -------|*3 /   - + ------- |       ||- - - -------|*3 /   - + ------- |     |                ||- - - -------|*3 /   - + ------- |   \/ 3 *sin|--||     |    ||- - + -------|*3 /   - + ------- |   \/ 3 *sin|--|          |   2*\/ 3 *cos|--|
     /  1   I*\/ 3              2                 \\  2      2   / \/    6      18   /       \\  2      2   / \/    6      18   /     |     /pi\       \\  2      2   / \/    6      18   /            \18/|     |    \\  2      2   / \/    6      18   /            \18/      /pi\|              \18/
2*3 /   - + -------  + -------------------- + ---------------------------------------- + ---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| + I*|---------------------------------------- - ------------- + cos|--|| - ---------------
  \/    6      18             _____________                      3                                          3                         \     \18/                      3                             3      /     \                   3                             3            \18//          3       
                             /         ___                                                                                                                                                                                                                                                             
                            /  1   I*\/ 3                                                                                                                                                                                                                                                              
                       3*3 /   - + -------                                                                                                                                                                                                                                                             
                         \/    6      18
−23cos⁡(π18)3+2re⁡(1(−12+3i2)16+3i183)3+2re⁡(1(−12−3i2)16+3i183)3+2316+3i183+i(2im⁡(1(−12+3i2)16+3i183)3−3sin⁡(π18)3+cos⁡(π18))+i(−cos⁡(π18)−3sin⁡(π18)3+2im⁡(1(−12−3i2)16+3i183)3)+216+3i183- \frac{2 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right) + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right) + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}} 
producto
/    /                1                 \     /                /                1                 \                \                          \ /                /                1                 \     /    /                1                 \                          \                \                                              
|2*re|----------------------------------|     |            2*im|----------------------------------|                |                          | |            2*re|----------------------------------|     |2*im|----------------------------------|                          |                |                                              
|    |                     _____________|     |                |                     _____________|                |                          | |                |                     _____________|     |    |                     _____________|                          |                |                                              
|    |/          ___\     /         ___ |     |                |/          ___\     /         ___ |                |                          | |                |/          ___\     /         ___ |     |    |/          ___\     /         ___ |                          |                |                                              
|    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |                ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\|     ___    /pi\          | |                ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\          |     ___    /pi\| /       _____________                       \
|    ||- - - -------|*3 /   - + ------- |     |                ||- - - -------|*3 /   - + ------- |   \/ 3 *sin|--||   \/ 3 *cos|--|          | |                ||- - + -------|*3 /   - + ------- |     |    ||- - + -------|*3 /   - + ------- |   \/ 3 *sin|--|          |   \/ 3 *cos|--|| |      /         ___                        |
|    \\  2      2   / \/    6      18   /     |     /pi\       \\  2      2   / \/    6      18   /            \18/|            \18/      /pi\| |     /pi\       \\  2      2   / \/    6      18   /     |    \\  2      2   / \/    6      18   /            \18/      /pi\|            \18/| |     /  1   I*\/ 3              2          |
|---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--||*|- sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - -------------|*|2*3 /   - + -------  + --------------------|
\                   3                         \     \18/                      3                             3      /         3            \18// \     \18/                      3                         \                   3                             3            \18//         3      / |  \/    6      18             _____________|
                                                                                                                                                                                                                                                                                                |                             /         ___ |
                                                                                                                                                                                                                                                                                                |                            /  1   I*\/ 3  |
                                                                                                                                                                                                                                                                                                |                       3*3 /   - + ------- |
                                                                                                                                                                                                                                                                                                \                         \/    6      18   /
(−3cos⁡(π18)3+2re⁡(1(−12−3i2)16+3i183)3+sin⁡(π18)+i(−cos⁡(π18)−3sin⁡(π18)3+2im⁡(1(−12−3i2)16+3i183)3))(2re⁡(1(−12+3i2)16+3i183)3−3cos⁡(π18)3−sin⁡(π18)+i(2im⁡(1(−12+3i2)16+3i183)3−3sin⁡(π18)3+cos⁡(π18)))(2316+3i183+216+3i183)\left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right)\right) \left(\frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right)\right) \left(\frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}\right) 
=
                   /                               2/3\
/            /pi\\ |  3 ____   3 ____ /        ___\   |
|-12 + 24*cos|--||*\2*\/ 18  + \/ 12 *\3 + I*\/ 3 /   /
\            \9 //                                     
-------------------------------------------------------
                        _____________                  
                     3 /         ___                   
                  27*\/  3 + I*\/ 3
(−12+24cos⁡(π9))(2183+123(3+3i)23)273+3i3\frac{\left(-12 + 24 \cos{\left(\frac{\pi}{9} \right)}\right) \left(2 \sqrt[3]{18} + \sqrt[3]{12} \left(3 + \sqrt{3} i\right)^{\frac{2}{3}}\right)}{27 \sqrt[3]{3 + \sqrt{3} i}} 
(-12 + 24*cos(pi/9))*(2*18^(1/3) + 12^(1/3)*(3 + i*sqrt(3))^(2/3))/(27*(3 + i*sqrt(3))^(1/3))
Respuesta rápida [src] 
         /                1                 \     /                /                1                 \                \                          
     2*re|----------------------------------|     |            2*im|----------------------------------|                |                          
         |                     _____________|     |                |                     _____________|                |                          
         |/          ___\     /         ___ |     |                |/          ___\     /         ___ |                |                          
         ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |                ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\|     ___    /pi\          
         ||- - - -------|*3 /   - + ------- |     |                ||- - - -------|*3 /   - + ------- |   \/ 3 *sin|--||   \/ 3 *cos|--|          
         \\  2      2   / \/    6      18   /     |     /pi\       \\  2      2   / \/    6      18   /            \18/|            \18/      /pi\
x1 = ---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--|
                        3                         \     \18/                      3                             3      /         3            \18/
x1=−3cos⁡(π18)3+2re⁡(1(−12−3i2)16+3i183)3+sin⁡(π18)+i(−cos⁡(π18)−3sin⁡(π18)3+2im⁡(1(−12−3i2)16+3i183)3)x_{1} = - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right) 
                     /                1                 \     /    /                1                 \                          \                
                 2*re|----------------------------------|     |2*im|----------------------------------|                          |                
                     |                     _____________|     |    |                     _____________|                          |                
                     |/          ___\     /         ___ |     |    |/          ___\     /         ___ |                          |                
                     ||  1   I*\/ 3 |    /  1   I*\/ 3  |     |    ||  1   I*\/ 3 |    /  1   I*\/ 3  |     ___    /pi\          |     ___    /pi\
                     ||- - + -------|*3 /   - + ------- |     |    ||- - + -------|*3 /   - + ------- |   \/ 3 *sin|--|          |   \/ 3 *cos|--|
          /pi\       \\  2      2   / \/    6      18   /     |    \\  2      2   / \/    6      18   /            \18/      /pi\|            \18/
x2 = - sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - -------------
          \18/                      3                         \                   3                             3            \18//         3
x2=2re⁡(1(−12+3i2)16+3i183)3−3cos⁡(π18)3−sin⁡(π18)+i(2im⁡(1(−12+3i2)16+3i183)3−3sin⁡(π18)3+cos⁡(π18))x_{2} = \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right) 
            _____________                       
           /         ___                        
          /  1   I*\/ 3              2          
x3 = 2*3 /   - + -------  + --------------------
       \/    6      18             _____________
                                  /         ___ 
                                 /  1   I*\/ 3  
                            3*3 /   - + ------- 
                              \/    6      18
x3=2316+3i183+216+3i183x_{3} = \frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}} 
x3 = 2/(3*(1/6 + sqrt(3)*i/18)^(1/3)) + 2*(1/6 + sqrt(3)*i/18)^(1/3)
Respuesta numérica [src] 
x1 = -1.48445439793712
x2 = 2.27431608520652
x3 = -0.789861687269397
x3 = -0.789861687269397
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