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x^6=(3x-2)^3

x^6=(3x-2)^3 equation

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Numerical solution:

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The solution

You have entered [src] 
 6            3
x  = (3*x - 2)
x6=(3x−2)3x^{6} = \left(3 x - 2\right)^{3}
Simplify
The graph
02468-8-6-4-21210-50000005000000
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Plot the graph f2(x)
Rapid solution [src] 
x1 = 1
x1=1x_{1} = 1 
x2 = 2
x2=2x_{2} = 2 
             /                       /    /     ___\\\              /    /     ___\\
             |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
             |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|
       3     |  3*\/ 3               \      2       /|              \      2       /
x3 = - - + I*|- ------- + ---------------------------| + ---------------------------
       4     \     4                   2             /                2
x3=−34+2174sin⁡(atan⁡(173)2)2+i(−334+2174cos⁡(atan⁡(173)2)2)x_{3} = - \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right) 
             /                     /    /     ___\\\              /    /     ___\\
             |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
             |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|
       3     |3*\/ 3               \      2       /|              \      2       /
x4 = - - + I*|------- + ---------------------------| - ---------------------------
       4     \   4                   2             /                2
x4=−2174sin⁡(atan⁡(173)2)2−34+i(334+2174cos⁡(atan⁡(173)2)2)x_{4} = - \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(\frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right) 
             /                     /    /     ___\\\              /    /     ___\\
             |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
             |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|
       3     |3*\/ 3               \      2       /|              \      2       /
x5 = - - + I*|------- - ---------------------------| + ---------------------------
       4     \   4                   2             /                2
x5=−34+2174sin⁡(atan⁡(173)2)2+i(−2174cos⁡(atan⁡(173)2)2+334)x_{5} = - \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + \frac{3 \sqrt{3}}{4}\right) 
             /                       /    /     ___\\\              /    /     ___\\
             |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
             |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|
       3     |  3*\/ 3               \      2       /|              \      2       /
x6 = - - + I*|- ------- - ---------------------------| - ---------------------------
       4     \     4                   2             /                2
x6=−2174sin⁡(atan⁡(173)2)2−34+i(−2174cos⁡(atan⁡(173)2)2−334)x_{6} = - \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3 \sqrt{3}}{4}\right) 
x6 = -217^(1/4)*sin(atan(17*sqrt(3))/2)/2 - 3/4 + i*(-217^(1/4)*cos(atan(17*sqrt(3))/2)/2 - 3*sqrt(3)/4)
Sum and product of roots [src] 
sum
                /                       /    /     ___\\\              /    /     ___\\           /                     /    /     ___\\\              /    /     ___\\           /                     /    /     ___\\\              /    /     ___\\           /                       /    /     ___\\\              /    /     ___\\
                |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|           |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|           |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|           |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
                |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|           |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|           |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|           |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|
          3     |  3*\/ 3               \      2       /|              \      2       /     3     |3*\/ 3               \      2       /|              \      2       /     3     |3*\/ 3               \      2       /|              \      2       /     3     |  3*\/ 3               \      2       /|              \      2       /
1 + 2 + - - + I*|- ------- + ---------------------------| + --------------------------- + - - + I*|------- + ---------------------------| - --------------------------- + - - + I*|------- - ---------------------------| + --------------------------- + - - + I*|- ------- - ---------------------------| - ---------------------------
          4     \     4                   2             /                2                  4     \   4                   2             /                2                  4     \   4                   2             /                2                  4     \     4                   2             /                2
(−2174sin⁡(atan⁡(173)2)2−34+i(−2174cos⁡(atan⁡(173)2)2−334))+((−34+2174sin⁡(atan⁡(173)2)2+i(−2174cos⁡(atan⁡(173)2)2+334))+(((1+2)+(−34+2174sin⁡(atan⁡(173)2)2+i(−334+2174cos⁡(atan⁡(173)2)2)))+(−2174sin⁡(atan⁡(173)2)2−34+i(334+2174cos⁡(atan⁡(173)2)2))))\left(- \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3 \sqrt{3}}{4}\right)\right) + \left(\left(- \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + \frac{3 \sqrt{3}}{4}\right)\right) + \left(\left(\left(1 + 2\right) + \left(- \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right)\right)\right) + \left(- \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(\frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right)\right)\right)\right) 
=
  /                       /    /     ___\\\     /                       /    /     ___\\\     /                     /    /     ___\\\     /                     /    /     ___\\\
  |            4 _____    |atan\17*\/ 3 /||     |            4 _____    |atan\17*\/ 3 /||     |          4 _____    |atan\17*\/ 3 /||     |          4 _____    |atan\17*\/ 3 /||
  |      ___   \/ 217 *cos|--------------||     |      ___   \/ 217 *cos|--------------||     |    ___   \/ 217 *cos|--------------||     |    ___   \/ 217 *cos|--------------||
  |  3*\/ 3               \      2       /|     |  3*\/ 3               \      2       /|     |3*\/ 3               \      2       /|     |3*\/ 3               \      2       /|
I*|- ------- + ---------------------------| + I*|- ------- - ---------------------------| + I*|------- + ---------------------------| + I*|------- - ---------------------------|
  \     4                   2             /     \     4                   2             /     \   4                   2             /     \   4                   2             /
i(−2174cos⁡(atan⁡(173)2)2−334)+i(−2174cos⁡(atan⁡(173)2)2+334)+i(−334+2174cos⁡(atan⁡(173)2)2)+i(334+2174cos⁡(atan⁡(173)2)2)i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3 \sqrt{3}}{4}\right) + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + \frac{3 \sqrt{3}}{4}\right) + i \left(- \frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right) + i \left(\frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right) 
product
  /        /                       /    /     ___\\\              /    /     ___\\\ /        /                     /    /     ___\\\              /    /     ___\\\ /        /                     /    /     ___\\\              /    /     ___\\\ /        /                       /    /     ___\\\              /    /     ___\\\
  |        |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|| |        |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|| |        |          4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|| |        |            4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /||
  |        |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|| |        |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|| |        |    ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------|| |        |      ___   \/ 217 *cos|--------------||   \/ 217 *sin|--------------||
  |  3     |  3*\/ 3               \      2       /|              \      2       /| |  3     |3*\/ 3               \      2       /|              \      2       /| |  3     |3*\/ 3               \      2       /|              \      2       /| |  3     |  3*\/ 3               \      2       /|              \      2       /|
2*|- - + I*|- ------- + ---------------------------| + ---------------------------|*|- - + I*|------- + ---------------------------| - ---------------------------|*|- - + I*|------- - ---------------------------| + ---------------------------|*|- - + I*|- ------- - ---------------------------| - ---------------------------|
  \  4     \     4                   2             /                2             / \  4     \   4                   2             /                2             / \  4     \   4                   2             /                2             / \  4     \     4                   2             /                2             /
2(−34+2174sin⁡(atan⁡(173)2)2+i(−334+2174cos⁡(atan⁡(173)2)2))(−2174sin⁡(atan⁡(173)2)2−34+i(334+2174cos⁡(atan⁡(173)2)2))(−34+2174sin⁡(atan⁡(173)2)2+i(−2174cos⁡(atan⁡(173)2)2+334))(−2174sin⁡(atan⁡(173)2)2−34+i(−2174cos⁡(atan⁡(173)2)2−334))2 \left(- \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right)\right) \left(- \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(\frac{3 \sqrt{3}}{4} + \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2}\right)\right) \left(- \frac{3}{4} + \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} + \frac{3 \sqrt{3}}{4}\right)\right) \left(- \frac{\sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3}{4} + i \left(- \frac{\sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}}{2} - \frac{3 \sqrt{3}}{4}\right)\right) 
=
8
88 
8
Numerical answer [src] 
x1 = 2.0
x2 = 0.583740972914917 - 0.0807680339067363*i
x3 = 0.583740972914917 + 0.0807680339067363*i
x4 = -2.08374097291492 - 2.67884424526005*i
x5 = 1.0
x6 = -2.08374097291492 + 2.67884424526005*i
x6 = -2.08374097291492 + 2.67884424526005*i
The graph
x^6=(3x-2)^3 equation
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