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x^3-81(x+3)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3                 
x  - 81*(x + 3) = 0
x3−81(x+3)=0x^{3} - 81 \left(x + 3\right) = 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=−81q = -81
v=dav = \frac{d}{a}
v=−243v = -243
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=−81x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -81
x1x2x3=−243x_{1} x_{2} x_{3} = -243
Respuesta rápida [src] 
                                                       /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\
                                                  9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|
         /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/
x1 = 9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - ---------------
         |                     _______________|       2         |    |                     _______________|       2              2       |          2       
         |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                  
         ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                  
         ||- - - -------|*3 /   - + --------- |                 |    ||- - - -------|*3 /   - + --------- |                              |                  
         \\  2      2   / \/    2       2     /                 \    \\  2      2   / \/    2       2     /                              /
x1=−33cos⁡(π18)2+9re⁡(1(−12−3i2)92+33i23)+9sin⁡(π18)2+i(−9cos⁡(π18)2−33sin⁡(π18)2+9im⁡(1(−12−3i2)92+33i23))x_{1} = - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right) 
                                                       /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\
                                                  9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|
         /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/
x2 = 9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - ---------------
         |                     _______________|       2         |    |                     _______________|       2              2       |          2       
         |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                  
         ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                  
         ||- - + -------|*3 /   - + --------- |                 |    ||- - + -------|*3 /   - + --------- |                              |                  
         \\  2      2   / \/    2       2     /                 \    \\  2      2   / \/    2       2     /                              /
x2=9re⁡(1(−12+3i2)92+33i23)−33cos⁡(π18)2−9sin⁡(π18)2+i(9im⁡(1(−12+3i2)92+33i23)−33sin⁡(π18)2+9cos⁡(π18)2)x_{2} = 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right) 
            _______________                       
           /           ___                        
          /  9   3*I*\/ 3              9          
x3 = 3*3 /   - + ---------  + --------------------
       \/    2       2             _______________
                                  /           ___ 
                                 /  9   3*I*\/ 3  
                              3 /   - + --------- 
                              \/    2       2
x3=992+33i23+392+33i23x_{3} = \frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}} 
x3 = 9/(9/2 + 3*sqrt(3)*i/2)^(1/3) + 3*(9/2 + 3*sqrt(3)*i/2)^(1/3)
Suma y producto de raíces [src] 
suma
                                                  /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\                                                     /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\          _______________                       
                                             9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|                                                9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|         /           ___                        
    /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/       /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/        /  9   3*I*\/ 3              9          
9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - --------------- + 9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - --------------- + 3*3 /   - + ---------  + --------------------
    |                     _______________|       2         |    |                     _______________|       2              2       |          2              |                     _______________|       2         |    |                     _______________|       2              2       |          2            \/    2       2             _______________
    |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                         |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                                                  /           ___ 
    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                         ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                                                 /  9   3*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
((9re⁡(1(−12+3i2)92+33i23)−33cos⁡(π18)2−9sin⁡(π18)2+i(9im⁡(1(−12+3i2)92+33i23)−33sin⁡(π18)2+9cos⁡(π18)2))+(−33cos⁡(π18)2+9re⁡(1(−12−3i2)92+33i23)+9sin⁡(π18)2+i(−9cos⁡(π18)2−33sin⁡(π18)2+9im⁡(1(−12−3i2)92+33i23))))+(992+33i23+392+33i23)\left(\left(9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right)\right) + \left(- \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(\frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}\right) 
=
       _______________                                                                                                                      /                                                  /pi\       ___    /pi\\     /                                                  /pi\       ___    /pi\\                  
      /           ___                                                                                                                       |                                             9*cos|--|   3*\/ 3 *sin|--||     |                                             9*cos|--|   3*\/ 3 *sin|--||                  
     /  9   3*I*\/ 3              9                 /                 1                  \       /                 1                  \     |    /                 1                  \        \18/              \18/|     |    /                 1                  \        \18/              \18/|       ___    /pi\
3*3 /   - + ---------  + -------------------- + 9*re|------------------------------------| + 9*re|------------------------------------| + I*|9*im|------------------------------------| + --------- - ---------------| + I*|9*im|------------------------------------| - --------- - ---------------| - 3*\/ 3 *cos|--|
  \/    2       2             _______________       |                     _______________|       |                     _______________|     |    |                     _______________|       2              2       |     |    |                     _______________|       2              2       |              \18/
                             /           ___        |/          ___\     /           ___ |       |/          ___\     /           ___ |     |    |/          ___\     /           ___ |                              |     |    |/          ___\     /           ___ |                              |                  
                            /  9   3*I*\/ 3         ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |       ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |     |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |     |    ||  1   I*\/ 3 |    /  9   3*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     /                              /
−33cos⁡(π18)+9re⁡(1(−12+3i2)92+33i23)+9re⁡(1(−12−3i2)92+33i23)+992+33i23+i(−9cos⁡(π18)2−33sin⁡(π18)2+9im⁡(1(−12−3i2)92+33i23))+i(9im⁡(1(−12+3i2)92+33i23)−33sin⁡(π18)2+9cos⁡(π18)2)+392+33i23- 3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right) + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right) + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}} 
producto
/                                                  /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\\ /                                                  /pi\     /                                                  /pi\       ___    /pi\\       ___    /pi\\ /       _______________                       \
|                                             9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|| |                                             9*sin|--|     |                                             9*cos|--|   3*\/ 3 *sin|--||   3*\/ 3 *cos|--|| |      /           ___                        |
|    /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/| |    /                 1                  \        \18/     |    /                 1                  \        \18/              \18/|              \18/| |     /  9   3*I*\/ 3              9          |
|9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - ---------------|*|9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - ---------------|*|3*3 /   - + ---------  + --------------------|
|    |                     _______________|       2         |    |                     _______________|       2              2       |          2       | |    |                     _______________|       2         |    |                     _______________|       2              2       |          2       | |  \/    2       2             _______________|
|    |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                  | |    |/          ___\     /           ___ |                 |    |/          ___\     /           ___ |                              |                  | |                             /           ___ |
|    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                  | |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                 |    ||  1   I*\/ 3 |    /  9   3*I*\/ 3  |                              |                  | |                            /  9   3*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     /
(−33cos⁡(π18)2+9re⁡(1(−12−3i2)92+33i23)+9sin⁡(π18)2+i(−9cos⁡(π18)2−33sin⁡(π18)2+9im⁡(1(−12−3i2)92+33i23)))(9re⁡(1(−12+3i2)92+33i23)−33cos⁡(π18)2−9sin⁡(π18)2+i(9im⁡(1(−12+3i2)92+33i23)−33sin⁡(π18)2+9cos⁡(π18)2))(992+33i23+392+33i23)\left(- \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right)\right) \left(9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right)\right) \left(\frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}\right) 
=
        /         /pi\\                                
        |      cos|--|| /                2/3          \
    2/3 |  1      \9 /| |/           ___\        3 ___|
54*3   *|- - + -------|*\\18 + 6*I*\/ 3 /    + 6*\/ 2 /
        \  4      2   /                                
-------------------------------------------------------
                       _____________                   
                    3 /         ___                    
                    \/  3 + I*\/ 3
54⋅323(−14+cos⁡(π9)2)(623+(18+63i)23)3+3i3\frac{54 \cdot 3^{\frac{2}{3}} \left(- \frac{1}{4} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2}\right) \left(6 \sqrt[3]{2} + \left(18 + 6 \sqrt{3} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{3 + \sqrt{3} i}} 
54*3^(2/3)*(-1/4 + cos(pi/9)/2)*((18 + 6*i*sqrt(3))^(2/3) + 6*2^(1/3))/(3 + i*sqrt(3))^(1/3)
Respuesta numérica [src] 
x1 = 10.2344223834293
x2 = -3.55437759271229
x3 = -6.68004479071703
x3 = -6.68004479071703
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