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x^4+2*x^2-8*x-7=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
 4      2              
x  + 2*x  - 8*x - 7 = 0
$$\left(- 8 x + \left(x^{4} + 2 x^{2}\right)\right) - 7 = 0$$
Gráfica
Suma y producto de raíces [src]
suma
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                                                                                 /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + -----------------------                                                                                     /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + -----------------------                                                                            I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - -----------------------                                                                            I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - ----------------------- 
          _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________              _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________                _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________                _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________ 
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       /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------            /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------              /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------              /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  
      /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3               /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3                 /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3                 /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      
     /       3     \/     27      3               ________________         /                                               /       3     \/     27      3               ________________                                    /       3     \/     27      3               ________________         /                                               /       3     \/     27      3               ________________                                      /       3     \/     27      3               ________________           /                                             /       3     \/     27      3               ________________                                      /       3     \/     27      3               ________________           /                                             /       3     \/     27      3               ________________                            
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  /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                                  /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                                    /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                                    /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                             
\/                                           \/     27      3         \/                                              \/                                           \/     27      3                                    \/                                           \/     27      3         \/                                              \/                                           \/     27      3                                      \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                      \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                 
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$$\left(\left(\left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2} + \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2}\right) + \left(\frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right)\right) + \left(- \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} - \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right)\right) + \left(- \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right)$$
=
0
$$0$$
producto
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|                                                                                 /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + ----------------------- | |                                                                                 /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + ----------------------- | |                                                                        I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - ----------------------- | |                                                                        I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - ----------------------- |
|          _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________ | |          _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________ | |            _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________ | |            _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________ |
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|       /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  | |       /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  | |         /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  | |         /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  |
|      /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      | |      /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      | |        /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      | |        /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      |
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|  /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                             | |  /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                             | |    /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                             | |    /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                             |
|\/                                           \/     27      3         \/                                              \/                                           \/     27      3                                 | |\/                                           \/     27      3         \/                                              \/                                           \/     27      3                                 | |  \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                 | |  \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                 |
|------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------|*|------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------|*|- ------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------|*|- ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------|
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$$\left(\frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right) \left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2} + \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2}\right) \left(- \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} - \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right) \left(- \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}\right)$$
=
-7
$$-7$$
-7
Respuesta rápida [src]
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                                                                                       /     8        /  172   4*\/ 29                                      16                                              40           
                                                                                      /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + ----------------------- 
               _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________ 
              /              ________________                                       /                                               /              ________________                                    /           ____  
             /              /           ____                                       /                                               /              /           ____                                    /  172   4*\/ 29   
            /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  
           /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      
          /       3     \/     27      3               ________________         /                                               /       3     \/     27      3               ________________                            
         /                                            /           ____         /                                               /                                            /           ____                             
        /                                            /  172   4*\/ 29         /                                               /                                            /  172   4*\/ 29                              
       /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                             
     \/                                           \/     27      3         \/                                              \/                                           \/     27      3                                 
x1 = ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------
                                      2                                                                                                          2                                                                       
$$x_{1} = \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}$$
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                                                                                       /     8        /  172   4*\/ 29                                      16                                              40           
                                                                                      /    - - - 2*3 /   --- + --------  + ------------------------------------------------------------------- + ----------------------- 
               _________________________________________________________             /       3     \/     27      3                  _________________________________________________________          ________________ 
              /              ________________                                       /                                               /              ________________                                    /           ____  
             /              /           ____                                       /                                               /              /           ____                                    /  172   4*\/ 29   
            /     4        /  172   4*\/ 29                40                     /                                               /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  
           /    - - + 2*3 /   --- + --------  - -----------------------          /                                               /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      
          /       3     \/     27      3               ________________         /                                               /       3     \/     27      3               ________________                            
         /                                            /           ____         /                                               /                                            /           ____                             
        /                                            /  172   4*\/ 29         /                                               /                                            /  172   4*\/ 29                              
       /                                        9*3 /   --- + --------       /                                               /                                        9*3 /   --- + --------                             
     \/                                           \/     27      3         \/                                              \/                                           \/     27      3                                 
x2 = ------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------
                                      2                                                                                                          2                                                                       
$$x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} - \frac{8}{3} + \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2} + \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2}$$
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                                                                                           /   8        /  172   4*\/ 29                                      16                                              40           
                                                                             I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - ----------------------- 
                 _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________ 
                /              ________________                                         /                                             /              ________________                                    /           ____  
               /              /           ____                                         /                                             /              /           ____                                    /  172   4*\/ 29   
              /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  
             /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      
            /       3     \/     27      3               ________________           /                                             /       3     \/     27      3               ________________                            
           /                                            /           ____           /                                             /                                            /           ____                             
          /                                            /  172   4*\/ 29           /                                             /                                            /  172   4*\/ 29                              
         /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                             
       \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                 
x3 = - ------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------
                                        2                                                                                                          2                                                                       
$$x_{3} = - \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} - \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}$$
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                                                                                             /            ________________                                                                                                 
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                                                                                           /   8        /  172   4*\/ 29                                      16                                              40           
                                                                             I*           /    - + 2*3 /   --- + --------  + ------------------------------------------------------------------- - ----------------------- 
                 _________________________________________________________               /     3     \/     27      3                  _________________________________________________________          ________________ 
                /              ________________                                         /                                             /              ________________                                    /           ____  
               /              /           ____                                         /                                             /              /           ____                                    /  172   4*\/ 29   
              /     4        /  172   4*\/ 29                40                       /                                             /     4        /  172   4*\/ 29                40              9*3 /   --- + --------  
             /    - - + 2*3 /   --- + --------  - -----------------------            /                                             /    - - + 2*3 /   --- + --------  - -----------------------      \/     27      3      
            /       3     \/     27      3               ________________           /                                             /       3     \/     27      3               ________________                            
           /                                            /           ____           /                                             /                                            /           ____                             
          /                                            /  172   4*\/ 29           /                                             /                                            /  172   4*\/ 29                              
         /                                        9*3 /   --- + --------         /                                             /                                        9*3 /   --- + --------                             
       \/                                           \/     27      3           \/                                            \/                                           \/     27      3                                 
x4 = - ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------
                                        2                                                                                                          2                                                                       
$$x_{4} = - \frac{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}{2} + \frac{i \sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} + \frac{8}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}} + \frac{16}{\sqrt{- \frac{40}{9 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}} - \frac{4}{3} + 2 \sqrt[3]{\frac{172}{27} + \frac{4 \sqrt{29}}{3}}}}}}{2}$$
x4 = -sqrt(-40/(9*(172/27 + 4*sqrt(29)/3)^(1/3)) - 4/3 + 2*(172/27 + 4*sqrt(29)/3)^(1/3))/2 + i*sqrt(-40/(9*(172/27 + 4*sqrt(29)/3)^(1/3)) + 8/3 + 2*(172/27 + 4*sqrt(29)/3)^(1/3) + 16/sqrt(-40/(9*(172/27 + 4*sqrt(29)/3)^(1/3)) - 4/3 + 2*(172/27 + 4*sqrt(29)/3)^(1/3)))/2
Respuesta numérica [src]
x1 = -0.626615337908865 + 2.14112063462735*i
x2 = -0.626615337908865 - 2.14112063462735*i
x3 = 1.96792216657326
x4 = -0.714691490755533
x4 = -0.714691490755533