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sqrt(3)*x^3+3*x^2-3*sqrt(3)*x-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
  ___  3      2       ___          
\/ 3 *x  + 3*x  - 3*\/ 3 *x - 1 = 0
(−33x+(3x3+3x2))−1=0\left(- 3 \sqrt{3} x + \left(\sqrt{3} x^{3} + 3 x^{2}\right)\right) - 1 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(−33x+(3x3+3x2))−1=0\left(- 3 \sqrt{3} x + \left(\sqrt{3} x^{3} + 3 x^{2}\right)\right) - 1 = 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
3(3x3+3x2−33x−1)3=0\frac{\sqrt{3} \left(\sqrt{3} x^{3} + 3 x^{2} - 3 \sqrt{3} x - 1\right)}{3} = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=3p = \sqrt{3}
q=caq = \frac{c}{a}
q=−3q = -3
v=dav = \frac{d}{a}
v=−33v = - \frac{\sqrt{3}}{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=−3x_{1} + x_{2} + x_{3} = - \sqrt{3}
x1x2+x1x3+x2x3=−3x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3
x1x2x3=−33x_{1} x_{2} x_{3} = - \frac{\sqrt{3}}{3}
Gráfica
05-15-10-51015-50005000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
                                                                        /                                                 ___    /pi\          \     ___    /pi\
                                                                ___     |                                               \/ 3 *sin|--|          |   \/ 3 *cos|--|
          /pi\       /                 1                  \   \/ 3      |      /                 1                  \            \9 /      /pi\|            \9 /
x1 = - sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + -------------
          \9 /       |/          ___\    _________________|     3       |      |/          ___\    _________________|         3            \9 /|         3      
                     ||  1   I*\/ 3 | 3 /      ___        |             |      ||  1   I*\/ 3 | 3 /      ___        |                          |                
                     ||- - - -------|*\/  12*\/ 3  + 36*I |             |      ||- - - -------|*\/  12*\/ 3  + 36*I |                          |                
                     \\  2      2   /                     /             \      \\  2      2   /                     /                          /
x1=−33−sin⁡(π9)−4re⁡(1(−12−3i2)123+36i3)+3cos⁡(π9)3+i(−4im⁡(1(−12−3i2)123+36i3)+3sin⁡(π9)3+cos⁡(π9))x_{1} = - \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right) 
                                                              /                                                           ___    /pi\\     ___    /pi\          
                                                      ___     |                                                         \/ 3 *sin|--||   \/ 3 *cos|--|          
           /                 1                  \   \/ 3      |     /pi\       /                 1                  \            \9 /|            \9 /      /pi\
x2 = - 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--|
           |/          ___\    _________________|     3       |     \9 /       |/          ___\    _________________|         3      |         3            \9 /
           ||  1   I*\/ 3 | 3 /      ___        |             |                ||  1   I*\/ 3 | 3 /      ___        |                |                          
           ||- - + -------|*\/  12*\/ 3  + 36*I |             |                ||- - + -------|*\/  12*\/ 3  + 36*I |                |                          
           \\  2      2   /                     /             \                \\  2      2   /                     /                /
x2=−33+sin⁡(π9)+3cos⁡(π9)3−4re⁡(1(−12+3i2)123+36i3)+i(−cos⁡(π9)+3sin⁡(π9)3−4im⁡(1(−12+3i2)123+36i3))x_{2} = - \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right) 
                                         _________________
                                ___   3 /      ___        
                4             \/ 3    \/  12*\/ 3  + 36*I 
x3 = - -------------------- - ----- - --------------------
          _________________     3              3          
       3 /      ___                                       
       \/  12*\/ 3  + 36*I
x3=−33−123+36i33−4123+36i3x_{3} = - \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}} 
x3 = -sqrt(3)/3 - (12*sqrt(3) + 36*i)^(1/3)/3 - 4/(12*sqrt(3) + 36*i)^(1/3)
Suma y producto de raíces [src] 
suma
                                                                   /                                                 ___    /pi\          \     ___    /pi\                                                            /                                                           ___    /pi\\     ___    /pi\                                                 _________________
                                                           ___     |                                               \/ 3 *sin|--|          |   \/ 3 *cos|--|                                                    ___     |                                                         \/ 3 *sin|--||   \/ 3 *cos|--|                                        ___   3 /      ___        
     /pi\       /                 1                  \   \/ 3      |      /                 1                  \            \9 /      /pi\|            \9 /         /                 1                  \   \/ 3      |     /pi\       /                 1                  \            \9 /|            \9 /      /pi\              4             \/ 3    \/  12*\/ 3  + 36*I 
- sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + ------------- + - 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--| + - -------------------- - ----- - --------------------
     \9 /       |/          ___\    _________________|     3       |      |/          ___\    _________________|         3            \9 /|         3               |/          ___\    _________________|     3       |     \9 /       |/          ___\    _________________|         3      |         3            \9 /        _________________     3              3          
                ||  1   I*\/ 3 | 3 /      ___        |             |      ||  1   I*\/ 3 | 3 /      ___        |                          |                         ||  1   I*\/ 3 | 3 /      ___        |             |                ||  1   I*\/ 3 | 3 /      ___        |                |                               3 /      ___                                       
                ||- - - -------|*\/  12*\/ 3  + 36*I |             |      ||- - - -------|*\/  12*\/ 3  + 36*I |                          |                         ||- - + -------|*\/  12*\/ 3  + 36*I |             |                ||- - + -------|*\/  12*\/ 3  + 36*I |                |                               \/  12*\/ 3  + 36*I                                
                \\  2      2   /                     /             \      \\  2      2   /                     /                          /                         \\  2      2   /                     /             \                \\  2      2   /                     /                /
((−33+sin⁡(π9)+3cos⁡(π9)3−4re⁡(1(−12+3i2)123+36i3)+i(−cos⁡(π9)+3sin⁡(π9)3−4im⁡(1(−12+3i2)123+36i3)))+(−33−sin⁡(π9)−4re⁡(1(−12−3i2)123+36i3)+3cos⁡(π9)3+i(−4im⁡(1(−12−3i2)123+36i3)+3sin⁡(π9)3+cos⁡(π9))))+(−33−123+36i33−4123+36i3)\left(\left(- \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right)\right) + \left(- \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right)\right)\right) + \left(- \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}\right) 
=
                                                                                                                              _________________     /                                                           ___    /pi\\     /                                                 ___    /pi\          \       ___    /pi\
                                                                                                                           3 /      ___             |                                                         \/ 3 *sin|--||     |                                               \/ 3 *sin|--|          |   2*\/ 3 *cos|--|
    ___            4                 /                 1                  \       /                 1                  \   \/  12*\/ 3  + 36*I      |     /pi\       /                 1                  \            \9 /|     |      /                 1                  \            \9 /      /pi\|              \9 /
- \/ 3  - -------------------- - 4*re|------------------------------------| - 4*re|------------------------------------| - -------------------- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + ---------------
             _________________       |/          ___\    _________________|       |/          ___\    _________________|            3               |     \9 /       |/          ___\    _________________|         3      |     |      |/          ___\    _________________|         3            \9 /|          3       
          3 /      ___               ||  1   I*\/ 3 | 3 /      ___        |       ||  1   I*\/ 3 | 3 /      ___        |                            |                ||  1   I*\/ 3 | 3 /      ___        |                |     |      ||  1   I*\/ 3 | 3 /      ___        |                          |                  
          \/  12*\/ 3  + 36*I        ||- - + -------|*\/  12*\/ 3  + 36*I |       ||- - - -------|*\/  12*\/ 3  + 36*I |                            |                ||- - + -------|*\/  12*\/ 3  + 36*I |                |     |      ||- - - -------|*\/  12*\/ 3  + 36*I |                          |                  
                                     \\  2      2   /                     /       \\  2      2   /                     /                            \                \\  2      2   /                     /                /     \      \\  2      2   /                     /                          /
−3−4re⁡(1(−12−3i2)123+36i3)−4re⁡(1(−12+3i2)123+36i3)+23cos⁡(π9)3−123+36i33+i(−4im⁡(1(−12−3i2)123+36i3)+3sin⁡(π9)3+cos⁡(π9))+i(−cos⁡(π9)+3sin⁡(π9)3−4im⁡(1(−12+3i2)123+36i3))−4123+36i3- \sqrt{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{2 \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right) + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right) - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}} 
producto
/                                                                   /                                                 ___    /pi\          \     ___    /pi\\ /                                                         /                                                           ___    /pi\\     ___    /pi\          \ /                                    _________________\
|                                                           ___     |                                               \/ 3 *sin|--|          |   \/ 3 *cos|--|| |                                                 ___     |                                                         \/ 3 *sin|--||   \/ 3 *cos|--|          | |                           ___   3 /      ___        |
|     /pi\       /                 1                  \   \/ 3      |      /                 1                  \            \9 /      /pi\|            \9 /| |      /                 1                  \   \/ 3      |     /pi\       /                 1                  \            \9 /|            \9 /      /pi\| |           4             \/ 3    \/  12*\/ 3  + 36*I |
|- sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + -------------|*|- 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--||*|- -------------------- - ----- - --------------------|
|     \9 /       |/          ___\    _________________|     3       |      |/          ___\    _________________|         3            \9 /|         3      | |      |/          ___\    _________________|     3       |     \9 /       |/          ___\    _________________|         3      |         3            \9 /| |     _________________     3              3          |
|                ||  1   I*\/ 3 | 3 /      ___        |             |      ||  1   I*\/ 3 | 3 /      ___        |                          |                | |      ||  1   I*\/ 3 | 3 /      ___        |             |                ||  1   I*\/ 3 | 3 /      ___        |                |                          | |  3 /      ___                                       |
|                ||- - - -------|*\/  12*\/ 3  + 36*I |             |      ||- - - -------|*\/  12*\/ 3  + 36*I |                          |                | |      ||- - + -------|*\/  12*\/ 3  + 36*I |             |                ||- - + -------|*\/  12*\/ 3  + 36*I |                |                          | \  \/  12*\/ 3  + 36*I                                /
\                \\  2      2   /                     /             \      \\  2      2   /                     /                          /                / \      \\  2      2   /                     /             \                \\  2      2   /                     /                /                          /
(−33−sin⁡(π9)−4re⁡(1(−12−3i2)123+36i3)+3cos⁡(π9)3+i(−4im⁡(1(−12−3i2)123+36i3)+3sin⁡(π9)3+cos⁡(π9)))(−33+sin⁡(π9)+3cos⁡(π9)3−4re⁡(1(−12+3i2)123+36i3)+i(−cos⁡(π9)+3sin⁡(π9)3−4im⁡(1(−12+3i2)123+36i3)))(−33−123+36i33−4123+36i3)\left(- \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(- \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right)\right) \left(- \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}\right) 
=
       /        _________________ /           _________________\\                                
3 ____ |     3 /      ___         |  ___   3 /      ___        || /          /2*pi\         /pi\\
\/ 18 *\12 + \/  12*\/ 3  + 36*I *\\/ 3  + \/  12*\/ 3  + 36*I //*|9 - 24*cos|----| + 12*cos|--||
                                                                  \          \ 9  /         \9 //
-------------------------------------------------------------------------------------------------
                                              _____________                                      
                                           3 /   ___                                             
                                       162*\/  \/ 3  + 3*I
183(12+(3+123+36i3)123+36i3)(−24cos⁡(2π9)+9+12cos⁡(π9))1623+3i3\frac{\sqrt[3]{18} \left(12 + \left(\sqrt{3} + \sqrt[3]{12 \sqrt{3} + 36 i}\right) \sqrt[3]{12 \sqrt{3} + 36 i}\right) \left(- 24 \cos{\left(\frac{2 \pi}{9} \right)} + 9 + 12 \cos{\left(\frac{\pi}{9} \right)}\right)}{162 \sqrt[3]{\sqrt{3} + 3 i}} 
18^(1/3)*(12 + (12*sqrt(3) + 36*i)^(1/3)*(sqrt(3) + (12*sqrt(3) + 36*i)^(1/3)))*(9 - 24*cos(2*pi/9) + 12*cos(pi/9))/(162*(sqrt(3) + 3*i)^(1/3))
Respuesta numérica [src] 
x1 = -2.74747741945462
x2 = -0.176326980708465
x3 = 1.19175359259421
x3 = 1.19175359259421
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