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x^3+3*x^2-1=0

x^3+3*x^2-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3      2        
x  + 3*x  - 1 = 0
(x3+3x2)−1=0\left(x^{3} + 3 x^{2}\right) - 1 = 0
Simplificar
Teorema de Cardano-Vieta
es ecuación cúbica reducida
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=3p = 3
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−1v = -1
Fórmulas de Cardano-Vieta
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=−3x_{1} + x_{2} + x_{3} = -3
x1x2+x1x3+x2x3=0x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0
x1x2x3=−1x_{1} x_{2} x_{3} = -1
Gráfica
05-15-10-51015-50005000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
             /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\
          cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|
             \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /
x1 = -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------
             2          |                     _________________|     |   2          |                     _________________|         2      |         2      
                        |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                
                        ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                
                        ||- - - -------|*3 /   -- + ---------- |     |              ||- - - -------|*3 /   -- + ---------- |                |                
                        \\  2      2   / \/    2        2      /     \              \\  2      2   / \/    2        2      /                /
x1=−1−3sin⁡(π9)2−3re⁡(1(−12−3i2)272+273i23)+cos⁡(π9)2+i(−3im⁡(1(−12−3i2)272+273i23)+sin⁡(π9)2+3cos⁡(π9)2)x_{1} = -1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) 
             /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\
          cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|
             \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /
x2 = -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------
             2          |                     _________________|     |   2          |                     _________________|         2      |         2      
                        |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                
                        ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                
                        ||- - + -------|*3 /   -- + ---------- |     |              ||- - + -------|*3 /   -- + ---------- |                |                
                        \\  2      2   / \/    2        2      /     \              \\  2      2   / \/    2        2      /                /
x2=−1+3sin⁡(π9)2+cos⁡(π9)2−3re⁡(1(−12+3i2)272+273i23)+i(−3cos⁡(π9)2+sin⁡(π9)2−3im⁡(1(−12+3i2)272+273i23))x_{2} = -1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right) 
                                        _________________
                                       /             ___ 
                                      /  27   27*I*\/ 3  
                                   3 /   -- + ---------- 
                    3              \/    2        2      
x3 = -1 - ---------------------- - ----------------------
               _________________             3           
              /             ___                          
             /  27   27*I*\/ 3                           
          3 /   -- + ----------                          
          \/    2        2
x3=−1−272+273i233−3272+273i23x_{3} = -1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}} 
x3 = -1 - (27/2 + 27*sqrt(3)*i/2)^(1/3)/3 - 3/(27/2 + 27*sqrt(3)*i/2)^(1/3)
Suma y producto de raíces [src] 
suma
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        /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\           /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\                                    /  27   27*I*\/ 3  
     cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|        cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|                                 3 /   -- + ---------- 
        \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /           \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /                  3              \/    2        2      
-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - ------------- + -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + ------------- + -1 - ---------------------- - ----------------------
        2          |                     _________________|     |   2          |                     _________________|         2      |         2                 2          |                     _________________|     |   2          |                     _________________|         2      |         2                   _________________             3           
                   |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                                      |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                            /             ___                          
                   ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                                      ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                           /  27   27*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
((−1+3sin⁡(π9)2+cos⁡(π9)2−3re⁡(1(−12+3i2)272+273i23)+i(−3cos⁡(π9)2+sin⁡(π9)2−3im⁡(1(−12+3i2)272+273i23)))+(−1−3sin⁡(π9)2−3re⁡(1(−12−3i2)272+273i23)+cos⁡(π9)2+i(−3im⁡(1(−12−3i2)272+273i23)+sin⁡(π9)2+3cos⁡(π9)2)))+(−1−272+273i233−3272+273i23)\left(\left(-1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) + \left(-1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right)\right) + \left(-1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right) 
=
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                                                                                                                               /  27   27*I*\/ 3       /   /pi\                                                    ___    /pi\\     /   /pi\                                                    ___    /pi\\          
                                                                                                                            3 /   -- + ----------      |sin|--|                                                  \/ 3 *cos|--||     |sin|--|                                                  \/ 3 *cos|--||          
               3                  /                  1                   \       /                  1                   \   \/    2        2           |   \9 /       /                  1                   \            \9 /|     |   \9 /       /                  1                   \            \9 /|      /pi\
-3 - ---------------------- - 3*re|--------------------------------------| - 3*re|--------------------------------------| - ---------------------- + I*|------- - 3*im|--------------------------------------| - -------------| + I*|------- - 3*im|--------------------------------------| + -------------| + cos|--|
          _________________       |                     _________________|       |                     _________________|             3                |   2          |                     _________________|         2      |     |   2          |                     _________________|         2      |      \9 /
         /             ___        |/          ___\     /             ___ |       |/          ___\     /             ___ |                              |              |/          ___\     /             ___ |                |     |              |/          ___\     /             ___ |                |          
        /  27   27*I*\/ 3         ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |       ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                              |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |     |              ||  1   I*\/ 3 |    /  27   27*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      /                /
−3−3re⁡(1(−12−3i2)272+273i23)−3re⁡(1(−12+3i2)272+273i23)+cos⁡(π9)−272+273i233+i(−3cos⁡(π9)2+sin⁡(π9)2−3im⁡(1(−12+3i2)272+273i23))+i(−3im⁡(1(−12−3i2)272+273i23)+sin⁡(π9)2+3cos⁡(π9)2)−3272+273i23-3 - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \cos{\left(\frac{\pi}{9} \right)} - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right) + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}} 
producto
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/        /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\\ /        /pi\                                                    /   /pi\                                                    ___    /pi\\     ___    /pi\\ |                                 /  27   27*I*\/ 3  |
|     cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|| |     cos|--|                                                    |sin|--|                                                  \/ 3 *cos|--||   \/ 3 *sin|--|| |                              3 /   -- + ---------- |
|        \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /| |        \9 /       /                  1                   \     |   \9 /       /                  1                   \            \9 /|            \9 /| |               3              \/    2        2      |
|-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------|*|-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------|*|-1 - ---------------------- - ----------------------|
|        2          |                     _________________|     |   2          |                     _________________|         2      |         2      | |        2          |                     _________________|     |   2          |                     _________________|         2      |         2      | |          _________________             3           |
|                   |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                | |                   |/          ___\     /             ___ |     |              |/          ___\     /             ___ |                |                | |         /             ___                          |
|                   ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                | |                   ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |     |              ||  1   I*\/ 3 |    /  27   27*I*\/ 3  |                |                | |        /  27   27*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                               /
(−1−3sin⁡(π9)2−3re⁡(1(−12−3i2)272+273i23)+cos⁡(π9)2+i(−3im⁡(1(−12−3i2)272+273i23)+sin⁡(π9)2+3cos⁡(π9)2))(−1+3sin⁡(π9)2+cos⁡(π9)2−3re⁡(1(−12+3i2)272+273i23)+i(−3cos⁡(π9)2+sin⁡(π9)2−3im⁡(1(−12+3i2)272+273i23)))(−1−272+273i233−3272+273i23)\left(-1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(-1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) \left(-1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right) 
=
                /             _____________ /            _____________\\                               
/         /pi\\ |  3 ___   3 /         ___  |     2/3 3 /         ___ || /       ___    /pi\      /pi\\
|2 - 4*sin|--||*\2*\/ 2  + \/  1 + I*\/ 3  *\2 + 2   *\/  1 + I*\/ 3  //*|-1 + \/ 3 *sin|--| + cos|--||
\         \18//                                                          \              \9 /      \9 //
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                                             3 /         ___                                           
                                           4*\/  1 + I*\/ 3
(2−4sin⁡(π18))(223+1+3i3(2+2231+3i3))(−1+3sin⁡(π9)+cos⁡(π9))41+3i3\frac{\left(2 - 4 \sin{\left(\frac{\pi}{18} \right)}\right) \left(2 \sqrt[3]{2} + \sqrt[3]{1 + \sqrt{3} i} \left(2 + 2^{\frac{2}{3}} \sqrt[3]{1 + \sqrt{3} i}\right)\right) \left(-1 + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)}\right)}{4 \sqrt[3]{1 + \sqrt{3} i}} 
(2 - 4*sin(pi/18))*(2*2^(1/3) + (1 + i*sqrt(3))^(1/3)*(2 + 2^(2/3)*(1 + i*sqrt(3))^(1/3)))*(-1 + sqrt(3)*sin(pi/9) + cos(pi/9))/(4*(1 + i*sqrt(3))^(1/3))
Respuesta numérica [src] 
x1 = 0.532088886237956
x2 = -2.87938524157182
x3 = -0.652703644666139
x3 = -0.652703644666139
Gráfico
x^3+3*x^2-1=0 la ecuación
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