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-x³+8y=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3          
- x  + 8*y = 0
−x3+8y=0- x^{3} + 8 y = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
−x3+8y=0- x^{3} + 8 y = 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
x3−8y=0x^{3} - 8 y = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−8yv = - 8 y
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=0x_{1} + x_{2} + x_{3} = 0
x1x2+x1x3+x2x3=0x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0
x1x2x3=−8yx_{1} x_{2} x_{3} = - 8 y
Gráfica
Suma y producto de raíces [src] 
suma
     _________________                                   _________________                              /     _________________                                     _________________                         \      _________________                                     _________________                              /     _________________                                     _________________                         \      _________________                                     _________________                         
  6 /   2        2        /atan2(im(y), re(y))\       6 /   2        2        /atan2(im(y), re(y))\     |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\     |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\
2*\/  im (y) + re (y) *cos|-------------------| + 2*I*\/  im (y) + re (y) *sin|-------------------| + I*|- \/  im (y) + re (y) *sin|-------------------| - \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| + \/ 3 *\/  im (y) + re (y) *sin|-------------------| + I*|- \/  im (y) + re (y) *sin|-------------------| + \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| - \/ 3 *\/  im (y) + re (y) *sin|-------------------|
                          \         3         /                               \         3         /     \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         /     \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         /
((2i(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+2(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+(i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3)))+(i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))−3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))\left(\left(2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right)\right) + \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) 
=
  /     _________________                                     _________________                         \     /     _________________                                     _________________                         \          _________________                         
  |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|     |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|       6 /   2        2        /atan2(im(y), re(y))\
I*|- \/  im (y) + re (y) *sin|-------------------| + \/ 3 *\/  im (y) + re (y) *cos|-------------------|| + I*|- \/  im (y) + re (y) *sin|-------------------| - \/ 3 *\/  im (y) + re (y) *cos|-------------------|| + 2*I*\/  im (y) + re (y) *sin|-------------------|
  \                          \         3         /                                 \         3         //     \                          \         3         /                                 \         3         //                               \         3         /
i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+2i(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + 2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} 
producto
/     _________________                                   _________________                         \ /  /     _________________                                     _________________                         \      _________________                                     _________________                         \ /  /     _________________                                     _________________                         \      _________________                                     _________________                         \
|  6 /   2        2        /atan2(im(y), re(y))\       6 /   2        2        /atan2(im(y), re(y))\| |  |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\| |  |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|
|2*\/  im (y) + re (y) *cos|-------------------| + 2*I*\/  im (y) + re (y) *sin|-------------------||*|I*|- \/  im (y) + re (y) *sin|-------------------| - \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| + \/ 3 *\/  im (y) + re (y) *sin|-------------------||*|I*|- \/  im (y) + re (y) *sin|-------------------| + \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| - \/ 3 *\/  im (y) + re (y) *sin|-------------------||
\                          \         3         /                               \         3         // \  \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         // \  \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         //
(2i(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+2(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))(i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))(i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))−3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))\left(2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) \left(i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) 
=
                       /                                                                                           /       im(y)              /atan2(im(y), re(y))\\                         \
                       |                                                                                       3*I*|-------------------- + sin|-------------------||                         |
                       |                                 /atan2(im(y), re(y))\                                     |   _________________      \         3         /|                         |
     _________________ |                            3*cos|-------------------|                                     |  /   2        2                               |                         |
    /   2        2     |   3/atan2(im(y), re(y))\        \         3         /        3/atan2(im(y), re(y))\       \\/  im (y) + re (y)                            /          3*re(y)        |
8*\/  im (y) + re (y) *|cos |-------------------| - -------------------------- - I*sin |-------------------| + ----------------------------------------------------- + ----------------------|
                       |    \         3         /               4                      \         3         /                             4                                  _________________|
                       |                                                                                                                                                   /   2        2    |
                       \                                                                                                                                               4*\/  im (y) + re (y) /
8(re⁡(y))2+(im⁡(y))2(3i(sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+im⁡(y)(re⁡(y))2+(im⁡(y))2)4−isin⁡3(atan2⁡(im⁡(y),re⁡(y))3)+cos⁡3(atan2⁡(im⁡(y),re⁡(y))3)−3cos⁡(atan2⁡(im⁡(y),re⁡(y))3)4+3re⁡(y)4(re⁡(y))2+(im⁡(y))2)8 \sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \frac{\operatorname{im}{\left(y\right)}}{\sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}}{4} + \frac{3 \operatorname{re}{\left(y\right)}}{4 \sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}\right) 
8*sqrt(im(y)^2 + re(y)^2)*(cos(atan2(im(y), re(y))/3)^3 - 3*cos(atan2(im(y), re(y))/3)/4 - i*sin(atan2(im(y), re(y))/3)^3 + 3*i*(im(y)/sqrt(im(y)^2 + re(y)^2) + sin(atan2(im(y), re(y))/3))/4 + 3*re(y)/(4*sqrt(im(y)^2 + re(y)^2)))
Respuesta rápida [src] 
          _________________                                   _________________                         
       6 /   2        2        /atan2(im(y), re(y))\       6 /   2        2        /atan2(im(y), re(y))\
x1 = 2*\/  im (y) + re (y) *cos|-------------------| + 2*I*\/  im (y) + re (y) *sin|-------------------|
                               \         3         /                               \         3         /
x1=2i(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+2(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3)x_{1} = 2 i \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + 2 \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} 
       /     _________________                                     _________________                         \      _________________                                     _________________                         
       |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\
x2 = I*|- \/  im (y) + re (y) *sin|-------------------| - \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| + \/ 3 *\/  im (y) + re (y) *sin|-------------------|
       \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         /
x2=i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))+3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3)x_{2} = i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} 
       /     _________________                                     _________________                         \      _________________                                     _________________                         
       |  6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\|   6 /   2        2        /atan2(im(y), re(y))\     ___ 6 /   2        2        /atan2(im(y), re(y))\
x3 = I*|- \/  im (y) + re (y) *sin|-------------------| + \/ 3 *\/  im (y) + re (y) *cos|-------------------|| - \/  im (y) + re (y) *cos|-------------------| - \/ 3 *\/  im (y) + re (y) *sin|-------------------|
       \                          \         3         /                                 \         3         //                           \         3         /                                 \         3         /
x3=i(−(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)+3(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3))−3(re⁡(y))2+(im⁡(y))26sin⁡(atan2⁡(im⁡(y),re⁡(y))3)−(re⁡(y))2+(im⁡(y))26cos⁡(atan2⁡(im⁡(y),re⁡(y))3)x_{3} = i \left(- \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} + \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)}\right) - \sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} - \sqrt[6]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{3} \right)} 
x3 = i*(-(re(y)^2 + im(y)^2)^(1/6)*sin(atan2(im(y, re(y))/3) + sqrt(3)*(re(y)^2 + im(y)^2)^(1/6)*cos(atan2(im(y), re(y))/3)) - sqrt(3)*(re(y)^2 + im(y)^2)^(1/6)*sin(atan2(im(y), re(y))/3) - (re(y)^2 + im(y)^2)^(1/6)*cos(atan2(im(y), re(y))/3))
    © Sr Examen – Калькулятор