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x^3+2x-4=0

x^3+2x-4=0 equation

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Numerical solution:

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The solution

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 3              
x  + 2*x - 4 = 0
(x3+2x)−4=0\left(x^{3} + 2 x\right) - 4 = 0
Simplify
Vieta's Theorem
it is reduced cubic equation
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=2q = 2
v=dav = \frac{d}{a}
v=−4v = -4
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=−4x_{1} x_{2} x_{3} = -4
The graph
-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.515.0-25002500
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            ______________                             /             ______________                        \
           /         ____                              |            /         ____                         |
          /      2*\/ 87                               |    ___    /      2*\/ 87                          |
       3 /   2 + --------                              |  \/ 3 *3 /   2 + --------              ___        |
       \/           9                  1               |        \/           9                \/ 3         |
x1 = - ------------------- + --------------------- + I*|- ------------------------- - ---------------------|
                2                   ______________     |              2                      ______________|
                                   /         ____      |                                    /         ____ |
                                  /      2*\/ 87       |                                   /      2*\/ 87  |
                             3*3 /   2 + --------      |                              3*3 /   2 + -------- |
                               \/           9          \                                \/           9     /
x1=−2+287932+132+28793+i(−32+287932−332+28793)x_{1} = - \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right) 
            ______________                             /           ______________                        \
           /         ____                              |          /         ____                         |
          /      2*\/ 87                               |  ___    /      2*\/ 87                          |
       3 /   2 + --------                              |\/ 3 *3 /   2 + --------              ___        |
       \/           9                  1               |      \/           9                \/ 3         |
x2 = - ------------------- + --------------------- + I*|------------------------- + ---------------------|
                2                   ______________     |            2                      ______________|
                                   /         ____      |                                  /         ____ |
                                  /      2*\/ 87       |                                 /      2*\/ 87  |
                             3*3 /   2 + --------      |                            3*3 /   2 + -------- |
                               \/           9          \                              \/           9     /
x2=−2+287932+132+28793+i(332+28793+32+287932)x_{2} = - \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right) 
          ______________                        
         /         ____                         
        /      2*\/ 87               2          
x3 = 3 /   2 + --------  - ---------------------
     \/           9               ______________
                                 /         ____ 
                                /      2*\/ 87  
                           3*3 /   2 + -------- 
                             \/           9
x3=−232+28793+2+28793x_{3} = - \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}} 
x3 = -2/(3*(2 + 2*sqrt(87)/9)^(1/3)) + (2 + 2*sqrt(87)/9)^(1/3)
Sum and product of roots [src] 
sum
       ______________                             /             ______________                        \          ______________                             /           ______________                        \                                              
      /         ____                              |            /         ____                         |         /         ____                              |          /         ____                         |                                              
     /      2*\/ 87                               |    ___    /      2*\/ 87                          |        /      2*\/ 87                               |  ___    /      2*\/ 87                          |        ______________                        
  3 /   2 + --------                              |  \/ 3 *3 /   2 + --------              ___        |     3 /   2 + --------                              |\/ 3 *3 /   2 + --------              ___        |       /         ____                         
  \/           9                  1               |        \/           9                \/ 3         |     \/           9                  1               |      \/           9                \/ 3         |      /      2*\/ 87               2          
- ------------------- + --------------------- + I*|- ------------------------- - ---------------------| + - ------------------- + --------------------- + I*|------------------------- + ---------------------| + 3 /   2 + --------  - ---------------------
           2                   ______________     |              2                      ______________|              2                   ______________     |            2                      ______________|   \/           9               ______________
                              /         ____      |                                    /         ____ |                                 /         ____      |                                  /         ____ |                               /         ____ 
                             /      2*\/ 87       |                                   /      2*\/ 87  |                                /      2*\/ 87       |                                 /      2*\/ 87  |                              /      2*\/ 87  
                        3*3 /   2 + --------      |                              3*3 /   2 + -------- |                           3*3 /   2 + --------      |                            3*3 /   2 + -------- |                         3*3 /   2 + -------- 
                          \/           9          \                                \/           9     /                             \/           9          \                              \/           9     /                           \/           9
(−232+28793+2+28793)+((−2+287932+132+28793+i(−32+287932−332+28793))+(−2+287932+132+28793+i(332+28793+32+287932)))\left(- \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)\right)\right) 
=
  /           ______________                        \     /             ______________                        \
  |          /         ____                         |     |            /         ____                         |
  |  ___    /      2*\/ 87                          |     |    ___    /      2*\/ 87                          |
  |\/ 3 *3 /   2 + --------              ___        |     |  \/ 3 *3 /   2 + --------              ___        |
  |      \/           9                \/ 3         |     |        \/           9                \/ 3         |
I*|------------------------- + ---------------------| + I*|- ------------------------- - ---------------------|
  |            2                      ______________|     |              2                      ______________|
  |                                  /         ____ |     |                                    /         ____ |
  |                                 /      2*\/ 87  |     |                                   /      2*\/ 87  |
  |                            3*3 /   2 + -------- |     |                              3*3 /   2 + -------- |
  \                              \/           9     /     \                                \/           9     /
i(−32+287932−332+28793)+i(332+28793+32+287932)i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right) + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right) 
product
/       ______________                             /             ______________                        \\ /       ______________                             /           ______________                        \\                                              
|      /         ____                              |            /         ____                         || |      /         ____                              |          /         ____                         ||                                              
|     /      2*\/ 87                               |    ___    /      2*\/ 87                          || |     /      2*\/ 87                               |  ___    /      2*\/ 87                          || /     ______________                        \
|  3 /   2 + --------                              |  \/ 3 *3 /   2 + --------              ___        || |  3 /   2 + --------                              |\/ 3 *3 /   2 + --------              ___        || |    /         ____                         |
|  \/           9                  1               |        \/           9                \/ 3         || |  \/           9                  1               |      \/           9                \/ 3         || |   /      2*\/ 87               2          |
|- ------------------- + --------------------- + I*|- ------------------------- - ---------------------||*|- ------------------- + --------------------- + I*|------------------------- + ---------------------||*|3 /   2 + --------  - ---------------------|
|           2                   ______________     |              2                      ______________|| |           2                   ______________     |            2                      ______________|| |\/           9               ______________|
|                              /         ____      |                                    /         ____ || |                              /         ____      |                                  /         ____ || |                            /         ____ |
|                             /      2*\/ 87       |                                   /      2*\/ 87  || |                             /      2*\/ 87       |                                 /      2*\/ 87  || |                           /      2*\/ 87  |
|                        3*3 /   2 + --------      |                              3*3 /   2 + -------- || |                        3*3 /   2 + --------      |                            3*3 /   2 + -------- || |                      3*3 /   2 + -------- |
\                          \/           9          \                                \/           9     // \                          \/           9          \                              \/           9     // \                        \/           9     /
(−2+287932+132+28793+i(332+28793+32+287932))(−2+287932+132+28793+i(−32+287932−332+28793))(−232+28793+2+28793)\left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right)\right) \left(- \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}\right) 
=
4
44 
4
Numerical answer [src] 
x1 = -0.589754512301458 - 1.74454325092266*i
x2 = 1.17950902460292
x3 = -0.589754512301458 + 1.74454325092266*i
x3 = -0.589754512301458 + 1.74454325092266*i
The graph
x^3+2x-4=0 equation
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