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-x^3+6*x^2+6*x+6=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3      2              
- x  + 6*x  + 6*x + 6 = 0
(6x+(−x3+6x2))+6=0\left(6 x + \left(- x^{3} + 6 x^{2}\right)\right) + 6 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(6x+(−x3+6x2))+6=0\left(6 x + \left(- x^{3} + 6 x^{2}\right)\right) + 6 = 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
x3−6x2−6x−6=0x^{3} - 6 x^{2} - 6 x - 6 = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=−6p = -6
q=caq = \frac{c}{a}
q=−6q = -6
v=dav = \frac{d}{a}
v=−6v = -6
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=6x_{1} + x_{2} + x_{3} = 6
x1x2+x1x3+x2x3=−6x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -6
x1x2x3=−6x_{1} x_{2} x_{3} = -6
Gráfica
-2.50.02.55.07.510.012.515.017.520.022.525.0-50005000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
                          _____________     /                            _____________\                             _____________     /         _____________                   \                                          
                       3 /        ____      |        ___          ___ 3 /        ____ |                          3 /        ____      |  ___ 3 /        ____            ___     |          _____________                   
           3           \/  17 + \/ 73       |    3*\/ 3         \/ 3 *\/  17 + \/ 73  |              3           \/  17 + \/ 73       |\/ 3 *\/  17 + \/ 73         3*\/ 3      |       3 /        ____           6        
2 - ---------------- - ---------------- + I*|---------------- - ----------------------| + 2 - ---------------- - ---------------- + I*|---------------------- - ----------------| + 2 + \/  17 + \/ 73   + ----------------
       _____________          2             |   _____________             2           |          _____________          2             |          2                 _____________|                             _____________
    3 /        ____                         |3 /        ____                          |       3 /        ____                         |                         3 /        ____ |                          3 /        ____ 
    \/  17 + \/ 73                          \\/  17 + \/ 73                           /       \/  17 + \/ 73                          \                         \/  17 + \/ 73  /                          \/  17 + \/ 73
(2+673+173+73+173)+((−73+1732−373+173+2+i(−373+1732+3373+173))+(−73+1732−373+173+2+i(−3373+173+373+1732)))\left(2 + \frac{6}{\sqrt[3]{\sqrt{73} + 17}} + \sqrt[3]{\sqrt{73} + 17}\right) + \left(\left(- \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2} + \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}}\right)\right) + \left(- \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2}\right)\right)\right) 
=
      /         _____________                   \     /                            _____________\
      |  ___ 3 /        ____            ___     |     |        ___          ___ 3 /        ____ |
      |\/ 3 *\/  17 + \/ 73         3*\/ 3      |     |    3*\/ 3         \/ 3 *\/  17 + \/ 73  |
6 + I*|---------------------- - ----------------| + I*|---------------- - ----------------------|
      |          2                 _____________|     |   _____________             2           |
      |                         3 /        ____ |     |3 /        ____                          |
      \                         \/  17 + \/ 73  /     \\/  17 + \/ 73                           /
6+i(−373+1732+3373+173)+i(−3373+173+373+1732)6 + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2} + \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}}\right) + i \left(- \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2}\right) 
producto
/                          _____________     /                            _____________\\ /                          _____________     /         _____________                   \\                                          
|                       3 /        ____      |        ___          ___ 3 /        ____ || |                       3 /        ____      |  ___ 3 /        ____            ___     || /       _____________                   \
|           3           \/  17 + \/ 73       |    3*\/ 3         \/ 3 *\/  17 + \/ 73  || |           3           \/  17 + \/ 73       |\/ 3 *\/  17 + \/ 73         3*\/ 3      || |    3 /        ____           6        |
|2 - ---------------- - ---------------- + I*|---------------- - ----------------------||*|2 - ---------------- - ---------------- + I*|---------------------- - ----------------||*|2 + \/  17 + \/ 73   + ----------------|
|       _____________          2             |   _____________             2           || |       _____________          2             |          2                 _____________|| |                          _____________|
|    3 /        ____                         |3 /        ____                          || |    3 /        ____                         |                         3 /        ____ || |                       3 /        ____ |
\    \/  17 + \/ 73                          \\/  17 + \/ 73                           // \    \/  17 + \/ 73                          \                         \/  17 + \/ 73  // \                       \/  17 + \/ 73  /
(−73+1732−373+173+2+i(−3373+173+373+1732))(−73+1732−373+173+2+i(−373+1732+3373+173))(2+673+173+73+173)\left(- \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2}\right)\right) \left(- \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2} + \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}}\right)\right) \left(2 + \frac{6}{\sqrt[3]{\sqrt{73} + 17}} + \sqrt[3]{\sqrt{73} + 17}\right) 
=
6
66 
6
Respuesta rápida [src] 
                               _____________     /                            _____________\
                            3 /        ____      |        ___          ___ 3 /        ____ |
                3           \/  17 + \/ 73       |    3*\/ 3         \/ 3 *\/  17 + \/ 73  |
x1 = 2 - ---------------- - ---------------- + I*|---------------- - ----------------------|
            _____________          2             |   _____________             2           |
         3 /        ____                         |3 /        ____                          |
         \/  17 + \/ 73                          \\/  17 + \/ 73                           /
x1=−73+1732−373+173+2+i(−373+1732+3373+173)x_{1} = - \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2} + \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}}\right) 
                               _____________     /         _____________                   \
                            3 /        ____      |  ___ 3 /        ____            ___     |
                3           \/  17 + \/ 73       |\/ 3 *\/  17 + \/ 73         3*\/ 3      |
x2 = 2 - ---------------- - ---------------- + I*|---------------------- - ----------------|
            _____________          2             |          2                 _____________|
         3 /        ____                         |                         3 /        ____ |
         \/  17 + \/ 73                          \                         \/  17 + \/ 73  /
x2=−73+1732−373+173+2+i(−3373+173+373+1732)x_{2} = - \frac{\sqrt[3]{\sqrt{73} + 17}}{2} - \frac{3}{\sqrt[3]{\sqrt{73} + 17}} + 2 + i \left(- \frac{3 \sqrt{3}}{\sqrt[3]{\sqrt{73} + 17}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{73} + 17}}{2}\right) 
            _____________                   
         3 /        ____           6        
x3 = 2 + \/  17 + \/ 73   + ----------------
                               _____________
                            3 /        ____ 
                            \/  17 + \/ 73
x3=2+673+173+73+173x_{3} = 2 + \frac{6}{\sqrt[3]{\sqrt{73} + 17}} + \sqrt[3]{\sqrt{73} + 17} 
x3 = 2 + 6/(sqrt(73) + 17)^(1/3) + (sqrt(73) + 17)^(1/3)
Respuesta numérica [src] 
x1 = 6.98237448409399
x2 = -0.491187242046996 - 0.786156234353676*i
x3 = -0.491187242046996 + 0.786156234353676*i
x3 = -0.491187242046996 + 0.786156234353676*i
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