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-x^3+12x^2-6x+8=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3       2              
- x  + 12*x  - 6*x + 8 = 0
(−6x+(−x3+12x2))+8=0\left(- 6 x + \left(- x^{3} + 12 x^{2}\right)\right) + 8 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(−6x+(−x3+12x2))+8=0\left(- 6 x + \left(- x^{3} + 12 x^{2}\right)\right) + 8 = 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
x3−12x2+6x−8=0x^{3} - 12 x^{2} + 6 x - 8 = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=−12p = -12
q=caq = \frac{c}{a}
q=6q = 6
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=12x_{1} + x_{2} + x_{3} = 12
x1x2+x1x3+x2x3=6x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 6
x1x2x3=−8x_{1} x_{2} x_{3} = -8
Gráfica
2345678910111213141516170250
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
                                 _______________     /                              _______________\
                              3 /           ___      |         ___           ___ 3 /           ___ |
                 7            \/  56 + 14*\/ 2       |     7*\/ 3          \/ 3 *\/  56 + 14*\/ 2  |
x1 = 4 - ------------------ - ------------------ + I*|------------------ - ------------------------|
            _______________           2              |   _______________              2            |
         3 /           ___                           |3 /           ___                            |
         \/  56 + 14*\/ 2                            \\/  56 + 14*\/ 2                             /
x1=−142+5632−7142+563+4+i(−3142+5632+73142+563)x_{1} = - \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right) 
                                 _______________     /         _______________                     \
                              3 /           ___      |  ___ 3 /           ___             ___      |
                 7            \/  56 + 14*\/ 2       |\/ 3 *\/  56 + 14*\/ 2          7*\/ 3       |
x2 = 4 - ------------------ - ------------------ + I*|------------------------ - ------------------|
            _______________           2              |           2                  _______________|
         3 /           ___                           |                           3 /           ___ |
         \/  56 + 14*\/ 2                            \                           \/  56 + 14*\/ 2  /
x2=−142+5632−7142+563+4+i(−73142+563+3142+5632)x_{2} = - \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right) 
            _______________                     
         3 /           ___            14        
x3 = 4 + \/  56 + 14*\/ 2   + ------------------
                                 _______________
                              3 /           ___ 
                              \/  56 + 14*\/ 2
x3=14142+563+4+142+563x_{3} = \frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56} 
x3 = 14/(14*sqrt(2) + 56)^(1/3) + 4 + (14*sqrt(2) + 56)^(1/3)
Suma y producto de raíces [src] 
suma
                            _______________     /                              _______________\                               _______________     /         _______________                     \                                              
                         3 /           ___      |         ___           ___ 3 /           ___ |                            3 /           ___      |  ___ 3 /           ___             ___      |          _______________                     
            7            \/  56 + 14*\/ 2       |     7*\/ 3          \/ 3 *\/  56 + 14*\/ 2  |               7            \/  56 + 14*\/ 2       |\/ 3 *\/  56 + 14*\/ 2          7*\/ 3       |       3 /           ___            14        
4 - ------------------ - ------------------ + I*|------------------ - ------------------------| + 4 - ------------------ - ------------------ + I*|------------------------ - ------------------| + 4 + \/  56 + 14*\/ 2   + ------------------
       _______________           2              |   _______________              2            |          _______________           2              |           2                  _______________|                               _______________
    3 /           ___                           |3 /           ___                            |       3 /           ___                           |                           3 /           ___ |                            3 /           ___ 
    \/  56 + 14*\/ 2                            \\/  56 + 14*\/ 2                             /       \/  56 + 14*\/ 2                            \                           \/  56 + 14*\/ 2  /                            \/  56 + 14*\/ 2
(14142+563+4+142+563)+((−142+5632−7142+563+4+i(−3142+5632+73142+563))+(−142+5632−7142+563+4+i(−73142+563+3142+5632)))\left(\frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56}\right) + \left(\left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right)\right) + \left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)\right)\right) 
=
       /         _______________                     \     /                              _______________\
       |  ___ 3 /           ___             ___      |     |         ___           ___ 3 /           ___ |
       |\/ 3 *\/  56 + 14*\/ 2          7*\/ 3       |     |     7*\/ 3          \/ 3 *\/  56 + 14*\/ 2  |
12 + I*|------------------------ - ------------------| + I*|------------------ - ------------------------|
       |           2                  _______________|     |   _______________              2            |
       |                           3 /           ___ |     |3 /           ___                            |
       \                           \/  56 + 14*\/ 2  /     \\/  56 + 14*\/ 2                             /
12+i(−3142+5632+73142+563)+i(−73142+563+3142+5632)12 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right) + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right) 
producto
/                            _______________     /                              _______________\\ /                            _______________     /         _______________                     \\                                              
|                         3 /           ___      |         ___           ___ 3 /           ___ || |                         3 /           ___      |  ___ 3 /           ___             ___      || /       _______________                     \
|            7            \/  56 + 14*\/ 2       |     7*\/ 3          \/ 3 *\/  56 + 14*\/ 2  || |            7            \/  56 + 14*\/ 2       |\/ 3 *\/  56 + 14*\/ 2          7*\/ 3       || |    3 /           ___            14        |
|4 - ------------------ - ------------------ + I*|------------------ - ------------------------||*|4 - ------------------ - ------------------ + I*|------------------------ - ------------------||*|4 + \/  56 + 14*\/ 2   + ------------------|
|       _______________           2              |   _______________              2            || |       _______________           2              |           2                  _______________|| |                            _______________|
|    3 /           ___                           |3 /           ___                            || |    3 /           ___                           |                           3 /           ___ || |                         3 /           ___ |
\    \/  56 + 14*\/ 2                            \\/  56 + 14*\/ 2                             // \    \/  56 + 14*\/ 2                            \                           \/  56 + 14*\/ 2  // \                         \/  56 + 14*\/ 2  /
(−142+5632−7142+563+4+i(−73142+563+3142+5632))(−142+5632−7142+563+4+i(−3142+5632+73142+563))(14142+563+4+142+563)\left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)\right) \left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right)\right) \left(\frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56}\right) 
=
8
88 
8
Respuesta numérica [src] 
x1 = 0.229926352383435 - 0.800228670667707*i
x2 = 0.229926352383435 + 0.800228670667707*i
x3 = 11.5401472952331
x3 = 11.5401472952331
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