Mister Exam

Other calculators

1/6x3^3-2x equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src] 
  3          
x3           
--- - 2*x = 0
 6
−2x+x336=0- 2 x + \frac{x_{3}^{3}}{6} = 0
Simplify
Vieta's Theorem
rewrite the equation
−2x+x336=0- 2 x + \frac{x_{3}^{3}}{6} = 0
of
ax33+bx32+cx3+d=0a x_{3}^{3} + b x_{3}^{2} + c x_{3} + d = 0
as reduced cubic equation
x33+bx32a+cx3a+da=0x_{3}^{3} + \frac{b x_{3}^{2}}{a} + \frac{c x_{3}}{a} + \frac{d}{a} = 0
−12x+x33=0- 12 x + x_{3}^{3} = 0
px32+qx3+v+x33=0p x_{3}^{2} + q x_{3} + v + x_{3}^{3} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−12xv = - 12 x
Vieta Formulas
x31+x32+x33=−px_{3 1} + x_{3 2} + x_{3 3} = - p
x31x32+x31x33+x32x33=qx_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = q
x31x32x33=vx_{3 1} x_{3 2} x_{3 3} = v
x31+x32+x33=0x_{3 1} + x_{3 2} + x_{3 3} = 0
x31x32+x31x33+x32x33=0x_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = 0
x31x32x33=−12xx_{3 1} x_{3 2} x_{3 3} = - 12 x
The graph
Rapid solution [src] 
                _________________                                        _________________                         
      3 ____ 6 /   2        2        /atan2(im(x), re(x))\     3 ____ 6 /   2        2        /atan2(im(x), re(x))\
x31 = \/ 12 *\/  im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/  im (x) + re (x) *sin|-------------------|
                                     \         3         /                                    \         3         /
x31=123i(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)+123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)x_{31} = \sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} 
        /            _________________                                         _________________                         \             _________________                                         _________________                         
        |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\
        |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------|
        |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /
x32 = I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + -------------------------------------------------------
        \                           2                                                        2                           /                            2                                                        2
x32=i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)+223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2x_{32} = i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} 
        /            _________________                                         _________________                         \             _________________                                         _________________                         
        |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\
        |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------|
        |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /
x33 = I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------
        \                           2                                                        2                           /                            2                                                        2
x33=i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2+223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)−223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2x_{33} = i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} 
x33 = i*(-12^(1/3)*(re(x)^2 + im(x)^2)^(1/6)*sin(atan2(im(x, re(x))/3)/2 + 2^(2/3)*3^(5/6)*(re(x)^2 + im(x)^2)^(1/6)*cos(atan2(im(x), re(x))/3)/2) - 2^(2/3)*3^(5/6)*(re(x)^2 + im(x)^2)^(1/6)*sin(atan2(im(x), re(x))/3)/2 - 12^(1/3)*(re(x)^2 + im(x)^2)^(1/6)*cos(atan2(im(x), re(x))/3)/2)
Sum and product of roots [src] 
sum
                                                                                                                  /            _________________                                         _________________                         \             _________________                                         _________________                              /            _________________                                         _________________                         \             _________________                                         _________________                         
                                                                                                                  |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\     |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\
          _________________                                        _________________                              |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------|     |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------|
3 ____ 6 /   2        2        /atan2(im(x), re(x))\     3 ____ 6 /   2        2        /atan2(im(x), re(x))\     |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /     |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /
\/ 12 *\/  im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/  im (x) + re (x) *sin|-------------------| + I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + ------------------------------------------------------- + I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------
                               \         3         /                                    \         3         /     \                           2                                                        2                           /                            2                                                        2                                \                           2                                                        2                           /                            2                                                        2
((123i(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)+123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3))+(i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)+223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2))+(i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2+223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)−223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)\left(\left(\sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}\right) + \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right)\right) + \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) 
=
  /            _________________                                         _________________                         \     /            _________________                                         _________________                         \                                                         
  |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|     |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|                                                         
  |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||     |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||               _________________                         
  |                                 \         3         /                                     \         3         /|     |                                 \         3         /                                     \         3         /|     3 ____ 6 /   2        2        /atan2(im(x), re(x))\
I*|- ---------------------------------------------------- + -------------------------------------------------------| + I*|- ---------------------------------------------------- - -------------------------------------------------------| + I*\/ 12 *\/  im (x) + re (x) *sin|-------------------|
  \                           2                                                        2                           /     \                           2                                                        2                           /                                    \         3         /
i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)+i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2+223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)+123i(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} 
product
                                                                                                                /  /            _________________                                         _________________                         \             _________________                                         _________________                         \ /  /            _________________                                         _________________                         \             _________________                                         _________________                         \
                                                                                                                |  |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\| |  |  3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|   3 ____ 6 /   2        2        /atan2(im(x), re(x))\    2/3  5/6 6 /   2        2        /atan2(im(x), re(x))\|
/          _________________                                        _________________                         \ |  |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------|| |  |  \/ 12 *\/  im (x) + re (x) *sin|-------------------|   2   *3   *\/  im (x) + re (x) *cos|-------------------||   \/ 12 *\/  im (x) + re (x) *cos|-------------------|   2   *3   *\/  im (x) + re (x) *sin|-------------------||
|3 ____ 6 /   2        2        /atan2(im(x), re(x))\     3 ____ 6 /   2        2        /atan2(im(x), re(x))\| |  |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /| |  |                                 \         3         /                                     \         3         /|                                  \         3         /                                     \         3         /|
|\/ 12 *\/  im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/  im (x) + re (x) *sin|-------------------||*|I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + -------------------------------------------------------|*|I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------|
\                               \         3         /                                    \         3         // \  \                           2                                                        2                           /                            2                                                        2                           / \  \                           2                                                        2                           /                            2                                                        2                           /
(123i(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)+123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3))(i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)+223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)(i(−123(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2+223⋅356(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)−223⋅356(re⁡(x))2+(im⁡(x))26sin⁡(atan2⁡(im⁡(x),re⁡(x))3)2−123(re⁡(x))2+(im⁡(x))26cos⁡(atan2⁡(im⁡(x),re⁡(x))3)2)\left(\sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}\right) \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) 
=
                        /                                                                                           /       im(x)              /atan2(im(x), re(x))\\                         \
                        |                                                                                       3*I*|-------------------- + sin|-------------------||                         |
                        |                                 /atan2(im(x), re(x))\                                     |   _________________      \         3         /|                         |
      _________________ |                            3*cos|-------------------|                                     |  /   2        2                               |                         |
     /   2        2     |   3/atan2(im(x), re(x))\        \         3         /        3/atan2(im(x), re(x))\       \\/  im (x) + re (x)                            /          3*re(x)        |
12*\/  im (x) + re (x) *|cos |-------------------| - -------------------------- - I*sin |-------------------| + ----------------------------------------------------- + ----------------------|
                        |    \         3         /               4                      \         3         /                             4                                  _________________|
                        |                                                                                                                                                   /   2        2    |
                        \                                                                                                                                               4*\/  im (x) + re (x) /
12(re⁡(x))2+(im⁡(x))2(3i(sin⁡(atan2⁡(im⁡(x),re⁡(x))3)+im⁡(x)(re⁡(x))2+(im⁡(x))2)4−isin⁡3(atan2⁡(im⁡(x),re⁡(x))3)+cos⁡3(atan2⁡(im⁡(x),re⁡(x))3)−3cos⁡(atan2⁡(im⁡(x),re⁡(x))3)4+3re⁡(x)4(re⁡(x))2+(im⁡(x))2)12 \sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \frac{\operatorname{im}{\left(x\right)}}{\sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{4} + \frac{3 \operatorname{re}{\left(x\right)}}{4 \sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}\right) 
12*sqrt(im(x)^2 + re(x)^2)*(cos(atan2(im(x), re(x))/3)^3 - 3*cos(atan2(im(x), re(x))/3)/4 - i*sin(atan2(im(x), re(x))/3)^3 + 3*i*(im(x)/sqrt(im(x)^2 + re(x)^2) + sin(atan2(im(x), re(x))/3))/4 + 3*re(x)/(4*sqrt(im(x)^2 + re(x)^2)))
    © Mister Exam – Calculator