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x^3+9*x^2+24*x+8=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3      2               
x  + 9*x  + 24*x + 8 = 0
(24x+(x3+9x2))+8=0\left(24 x + \left(x^{3} + 9 x^{2}\right)\right) + 8 = 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=9p = 9
q=caq = \frac{c}{a}
q=24q = 24
v=dav = \frac{d}{a}
v=8v = 8
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=−9x_{1} + x_{2} + x_{3} = -9
x1x2+x1x3+x2x3=24x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 24
x1x2x3=8x_{1} x_{2} x_{3} = 8
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-25002500
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
                                  _____________     /                              _____________\
                               3 /         ___      |        ___            ___ 3 /         ___ |
                  1            \/  5 + 2*\/ 6       |      \/ 3           \/ 3 *\/  5 + 2*\/ 6  |
x1 = -3 - ------------------ - ---------------- + I*|------------------ - ----------------------|
               _____________          2             |     _____________             2           |
            3 /         ___                         |  3 /         ___                          |
          2*\/  5 + 2*\/ 6                          \2*\/  5 + 2*\/ 6                           /
x1=−3−26+532−1226+53+i(−326+532+3226+53)x_{1} = -3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right) 
                                  _____________     /         _____________                     \
                               3 /         ___      |  ___ 3 /         ___            ___       |
                  1            \/  5 + 2*\/ 6       |\/ 3 *\/  5 + 2*\/ 6           \/ 3        |
x2 = -3 - ------------------ - ---------------- + I*|---------------------- - ------------------|
               _____________          2             |          2                   _____________|
            3 /         ___                         |                           3 /         ___ |
          2*\/  5 + 2*\/ 6                          \                         2*\/  5 + 2*\/ 6  /
x2=−3−26+532−1226+53+i(−3226+53+326+532)x_{2} = -3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right) 
                                _____________
                 1           3 /         ___ 
x3 = -3 + ---------------- + \/  5 + 2*\/ 6  
             _____________                   
          3 /         ___                    
          \/  5 + 2*\/ 6
x3=−3+126+53+26+53x_{3} = -3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5} 
x3 = -3 + (2*sqrt(6) + 5)^(-1/3) + (2*sqrt(6) + 5)^(1/3)
Suma y producto de raíces [src] 
suma
                             _____________     /                              _____________\                                _____________     /         _____________                     \                                           
                          3 /         ___      |        ___            ___ 3 /         ___ |                             3 /         ___      |  ___ 3 /         ___            ___       |                              _____________
             1            \/  5 + 2*\/ 6       |      \/ 3           \/ 3 *\/  5 + 2*\/ 6  |                1            \/  5 + 2*\/ 6       |\/ 3 *\/  5 + 2*\/ 6           \/ 3        |               1           3 /         ___ 
-3 - ------------------ - ---------------- + I*|------------------ - ----------------------| + -3 - ------------------ - ---------------- + I*|---------------------- - ------------------| + -3 + ---------------- + \/  5 + 2*\/ 6  
          _____________          2             |     _____________             2           |             _____________          2             |          2                   _____________|           _____________                   
       3 /         ___                         |  3 /         ___                          |          3 /         ___                         |                           3 /         ___ |        3 /         ___                    
     2*\/  5 + 2*\/ 6                          \2*\/  5 + 2*\/ 6                           /        2*\/  5 + 2*\/ 6                          \                         2*\/  5 + 2*\/ 6  /        \/  5 + 2*\/ 6
(−3+126+53+26+53)+((−3−26+532−1226+53+i(−326+532+3226+53))+(−3−26+532−1226+53+i(−3226+53+326+532)))\left(-3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5}\right) + \left(\left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right)\right) + \left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right)\right)\right) 
=
       /                              _____________\     /         _____________                     \
       |        ___            ___ 3 /         ___ |     |  ___ 3 /         ___            ___       |
       |      \/ 3           \/ 3 *\/  5 + 2*\/ 6  |     |\/ 3 *\/  5 + 2*\/ 6           \/ 3        |
-9 + I*|------------------ - ----------------------| + I*|---------------------- - ------------------|
       |     _____________             2           |     |          2                   _____________|
       |  3 /         ___                          |     |                           3 /         ___ |
       \2*\/  5 + 2*\/ 6                           /     \                         2*\/  5 + 2*\/ 6  /
−9+i(−326+532+3226+53)+i(−3226+53+326+532)-9 + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right) 
producto
/                             _____________     /                              _____________\\ /                             _____________     /         _____________                     \\                                           
|                          3 /         ___      |        ___            ___ 3 /         ___ || |                          3 /         ___      |  ___ 3 /         ___            ___       || /                           _____________\
|             1            \/  5 + 2*\/ 6       |      \/ 3           \/ 3 *\/  5 + 2*\/ 6  || |             1            \/  5 + 2*\/ 6       |\/ 3 *\/  5 + 2*\/ 6           \/ 3        || |            1           3 /         ___ |
|-3 - ------------------ - ---------------- + I*|------------------ - ----------------------||*|-3 - ------------------ - ---------------- + I*|---------------------- - ------------------||*|-3 + ---------------- + \/  5 + 2*\/ 6  |
|          _____________          2             |     _____________             2           || |          _____________          2             |          2                   _____________|| |        _____________                   |
|       3 /         ___                         |  3 /         ___                          || |       3 /         ___                         |                           3 /         ___ || |     3 /         ___                    |
\     2*\/  5 + 2*\/ 6                          \2*\/  5 + 2*\/ 6                           // \     2*\/  5 + 2*\/ 6                          \                         2*\/  5 + 2*\/ 6  // \     \/  5 + 2*\/ 6                     /
(−3−26+532−1226+53+i(−3226+53+326+532))(−3−26+532−1226+53+i(−326+532+3226+53))(−3+126+53+26+53)\left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right)\right) \left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right)\right) \left(-3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5}\right) 
=
-8
−8-8 
-8
Respuesta numérica [src] 
x1 = -0.387112135282455
x2 = -4.30644393235877 - 1.45615495232862*i
x3 = -4.30644393235877 + 1.45615495232862*i
x3 = -4.30644393235877 + 1.45615495232862*i
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