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x^3+2x+2=0

x^3+2x+2=0 equation

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Numerical solution:

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The solution

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 3              
x  + 2*x + 2 = 0
x3+2x+2=0x^{3} + 2 x + 2 = 0
Simplify
Vieta's Theorem
it is reduced cubic equation
px2+x3+qx+v=0p x^{2} + x^{3} + q x + v = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=2q = 2
v=dav = \frac{d}{a}
v=2v = 2
Vieta Formulas
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=2x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2
x1x2x3=2x_{1} x_{2} x_{3} = 2
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-25002500
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
                               ________________     /                               ________________\                              ________________     /                                 ________________\                            ________________
                            3 /          _____      |         ___            ___ 3 /          _____ |                           3 /          _____      |           ___            ___ 3 /          _____ |                         3 /          _____ 
               1            \/  27 + 3*\/ 105       |       \/ 3           \/ 3 *\/  27 + 3*\/ 105  |              1            \/  27 + 3*\/ 105       |         \/ 3           \/ 3 *\/  27 + 3*\/ 105  |            2            \/  27 + 3*\/ 105  
0 + - ------------------- + ------------------- + I*|------------------- + -------------------------| + - ------------------- + ------------------- + I*|- ------------------- - -------------------------| + ------------------- - -------------------
         ________________            6              |   ________________               6            |        ________________            6              |     ________________               6            |      ________________            3         
      3 /          _____                            |3 /          _____                             |     3 /          _____                            |  3 /          _____                             |   3 /          _____                       
      \/  27 + 3*\/ 105                             \\/  27 + 3*\/ 105                              /     \/  27 + 3*\/ 105                             \  \/  27 + 3*\/ 105                              /   \/  27 + 3*\/ 105
(−27+310533+227+31053)+((−127+31053+27+310536+i(−327+310536−327+31053))+(0+(−127+31053+27+310536+i(327+31053+327+310536))))\left(- \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}\right) + \left(\left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)\right) + \left(0 + \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)\right)\right)\right) 
=
  /                               ________________\     /                                 ________________\
  |         ___            ___ 3 /          _____ |     |           ___            ___ 3 /          _____ |
  |       \/ 3           \/ 3 *\/  27 + 3*\/ 105  |     |         \/ 3           \/ 3 *\/  27 + 3*\/ 105  |
I*|------------------- + -------------------------| + I*|- ------------------- - -------------------------|
  |   ________________               6            |     |     ________________               6            |
  |3 /          _____                             |     |  3 /          _____                             |
  \\/  27 + 3*\/ 105                              /     \  \/  27 + 3*\/ 105                              /
i(−327+310536−327+31053)+i(327+31053+327+310536)i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right) + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right) 
product
  /                           ________________     /                               ________________\\ /                           ________________     /                                 ________________\\ /                         ________________\
  |                        3 /          _____      |         ___            ___ 3 /          _____ || |                        3 /          _____      |           ___            ___ 3 /          _____ || |                      3 /          _____ |
  |           1            \/  27 + 3*\/ 105       |       \/ 3           \/ 3 *\/  27 + 3*\/ 105  || |           1            \/  27 + 3*\/ 105       |         \/ 3           \/ 3 *\/  27 + 3*\/ 105  || |         2            \/  27 + 3*\/ 105  |
1*|- ------------------- + ------------------- + I*|------------------- + -------------------------||*|- ------------------- + ------------------- + I*|- ------------------- - -------------------------||*|------------------- - -------------------|
  |     ________________            6              |   ________________               6            || |     ________________            6              |     ________________               6            || |   ________________            3         |
  |  3 /          _____                            |3 /          _____                             || |  3 /          _____                            |  3 /          _____                             || |3 /          _____                       |
  \  \/  27 + 3*\/ 105                             \\/  27 + 3*\/ 105                              // \  \/  27 + 3*\/ 105                             \  \/  27 + 3*\/ 105                              // \\/  27 + 3*\/ 105                        /
1(−127+31053+27+310536+i(327+31053+327+310536))(−127+31053+27+310536+i(−327+310536−327+31053))(−27+310533+227+31053)1 \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)\right) \left(- \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)\right) \left(- \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}\right) 
=
-2
−2-2 
-2
Rapid solution [src] 
                                ________________     /                               ________________\
                             3 /          _____      |         ___            ___ 3 /          _____ |
                1            \/  27 + 3*\/ 105       |       \/ 3           \/ 3 *\/  27 + 3*\/ 105  |
x1 = - ------------------- + ------------------- + I*|------------------- + -------------------------|
          ________________            6              |   ________________               6            |
       3 /          _____                            |3 /          _____                             |
       \/  27 + 3*\/ 105                             \\/  27 + 3*\/ 105                              /
x1=−127+31053+27+310536+i(327+31053+327+310536)x_{1} = - \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(\frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6}\right)
Simplify 
                                ________________     /                                 ________________\
                             3 /          _____      |           ___            ___ 3 /          _____ |
                1            \/  27 + 3*\/ 105       |         \/ 3           \/ 3 *\/  27 + 3*\/ 105  |
x2 = - ------------------- + ------------------- + I*|- ------------------- - -------------------------|
          ________________            6              |     ________________               6            |
       3 /          _____                            |  3 /          _____                             |
       \/  27 + 3*\/ 105                             \  \/  27 + 3*\/ 105                              /
x2=−127+31053+27+310536+i(−327+310536−327+31053)x_{2} = - \frac{1}{\sqrt[3]{27 + 3 \sqrt{105}}} + \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 3 \sqrt{105}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{27 + 3 \sqrt{105}}}\right)
Simplify 
                              ________________
                           3 /          _____ 
              2            \/  27 + 3*\/ 105  
x3 = ------------------- - -------------------
        ________________            3         
     3 /          _____                       
     \/  27 + 3*\/ 105
x3=−27+310533+227+31053x_{3} = - \frac{\sqrt[3]{27 + 3 \sqrt{105}}}{3} + \frac{2}{\sqrt[3]{27 + 3 \sqrt{105}}}
Simplify
Numerical answer [src] 
x1 = 0.385458498529624 + 1.56388451052696*i
x2 = 0.385458498529624 - 1.56388451052696*i
x3 = -0.770916997059248
x3 = -0.770916997059248
The graph
x^3+2x+2=0 equation
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