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x^3+12*x-20=0 equation

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Numerical solution:

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The solution

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 3                
x  + 12*x - 20 = 0
(x3+12x)−20=0\left(x^{3} + 12 x\right) - 20 = 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=12q = 12
v=dav = \frac{d}{a}
v=−20v = -20
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=12x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 12
x1x2x3=−20x_{1} x_{2} x_{3} = -20
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] 
                             _______________     /                                _______________\
                          3 /          ____      |           ___           ___ 3 /          ____ |
             2            \/  10 + 2*\/ 41       |       2*\/ 3          \/ 3 *\/  10 + 2*\/ 41  |
x1 = ------------------ - ------------------ + I*|- ------------------ - ------------------------|
        _______________           2              |     _______________              2            |
     3 /          ____                           |  3 /          ____                            |
     \/  10 + 2*\/ 41                            \  \/  10 + 2*\/ 41                             /
x1=−10+24132+210+2413+i(−310+24132−2310+2413)x_{1} = - \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right) 
                             _______________     /         _______________                     \
                          3 /          ____      |  ___ 3 /          ____             ___      |
             2            \/  10 + 2*\/ 41       |\/ 3 *\/  10 + 2*\/ 41          2*\/ 3       |
x2 = ------------------ - ------------------ + I*|------------------------ + ------------------|
        _______________           2              |           2                  _______________|
     3 /          ____                           |                           3 /          ____ |
     \/  10 + 2*\/ 41                            \                           \/  10 + 2*\/ 41  /
x2=−10+24132+210+2413+i(2310+2413+310+24132)x_{2} = - \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right) 
        _______________                     
     3 /          ____            4         
x3 = \/  10 + 2*\/ 41   - ------------------
                             _______________
                          3 /          ____ 
                          \/  10 + 2*\/ 41
x3=−410+2413+10+2413x_{3} = - \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}} 
x3 = -4/(10 + 2*sqrt(41))^(1/3) + (10 + 2*sqrt(41))^(1/3)
Sum and product of roots [src] 
sum
                        _______________     /                                _______________\                           _______________     /         _______________                     \                                          
                     3 /          ____      |           ___           ___ 3 /          ____ |                        3 /          ____      |  ___ 3 /          ____             ___      |      _______________                     
        2            \/  10 + 2*\/ 41       |       2*\/ 3          \/ 3 *\/  10 + 2*\/ 41  |           2            \/  10 + 2*\/ 41       |\/ 3 *\/  10 + 2*\/ 41          2*\/ 3       |   3 /          ____            4         
------------------ - ------------------ + I*|- ------------------ - ------------------------| + ------------------ - ------------------ + I*|------------------------ + ------------------| + \/  10 + 2*\/ 41   - ------------------
   _______________           2              |     _______________              2            |      _______________           2              |           2                  _______________|                           _______________
3 /          ____                           |  3 /          ____                            |   3 /          ____                           |                           3 /          ____ |                        3 /          ____ 
\/  10 + 2*\/ 41                            \  \/  10 + 2*\/ 41                             /   \/  10 + 2*\/ 41                            \                           \/  10 + 2*\/ 41  /                        \/  10 + 2*\/ 41
(−410+2413+10+2413)+((−10+24132+210+2413+i(−310+24132−2310+2413))+(−10+24132+210+2413+i(2310+2413+310+24132)))\left(- \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}}\right) + \left(\left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right)\right) + \left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)\right)\right) 
=
  /         _______________                     \     /                                _______________\
  |  ___ 3 /          ____             ___      |     |           ___           ___ 3 /          ____ |
  |\/ 3 *\/  10 + 2*\/ 41          2*\/ 3       |     |       2*\/ 3          \/ 3 *\/  10 + 2*\/ 41  |
I*|------------------------ + ------------------| + I*|- ------------------ - ------------------------|
  |           2                  _______________|     |     _______________              2            |
  |                           3 /          ____ |     |  3 /          ____                            |
  \                           \/  10 + 2*\/ 41  /     \  \/  10 + 2*\/ 41                             /
i(−310+24132−2310+2413)+i(2310+2413+310+24132)i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right) + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right) 
product
/                        _______________     /                                _______________\\ /                        _______________     /         _______________                     \\                                          
|                     3 /          ____      |           ___           ___ 3 /          ____ || |                     3 /          ____      |  ___ 3 /          ____             ___      || /   _______________                     \
|        2            \/  10 + 2*\/ 41       |       2*\/ 3          \/ 3 *\/  10 + 2*\/ 41  || |        2            \/  10 + 2*\/ 41       |\/ 3 *\/  10 + 2*\/ 41          2*\/ 3       || |3 /          ____            4         |
|------------------ - ------------------ + I*|- ------------------ - ------------------------||*|------------------ - ------------------ + I*|------------------------ + ------------------||*|\/  10 + 2*\/ 41   - ------------------|
|   _______________           2              |     _______________              2            || |   _______________           2              |           2                  _______________|| |                        _______________|
|3 /          ____                           |  3 /          ____                            || |3 /          ____                           |                           3 /          ____ || |                     3 /          ____ |
\\/  10 + 2*\/ 41                            \  \/  10 + 2*\/ 41                             // \\/  10 + 2*\/ 41                            \                           \/  10 + 2*\/ 41  // \                     \/  10 + 2*\/ 41  /
(−10+24132+210+2413+i(2310+2413+310+24132))(−10+24132+210+2413+i(−310+24132−2310+2413))(−410+2413+10+2413)\left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)\right) \left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right)\right) \left(- \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}}\right) 
=
20
2020 
20
Numerical answer [src] 
x1 = -0.712675737640377 - 3.67746109715164*i
x2 = 1.42535147528075
x3 = -0.712675737640377 + 3.67746109715164*i
x3 = -0.712675737640377 + 3.67746109715164*i
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