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x^6=(6x-8)^3

x^6=(6x-8)^3 equation

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Numerical solution:

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The solution

You have entered [src] 
 6            3
x  = (6*x - 8)
x6=(6x−8)3x^{6} = \left(6 x - 8\right)^{3}
Simplify
The graph
02468-8-6-4-2141012-50005000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x_1 = 2
x1=2x_{1} = 2
Simplify 
x_2 = 4
x2=4x_{2} = 4
Simplify 
              /      ___              /    /     ___\\\              /    /     ___\\
        3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
x_3 = - - + I*|- ------- + \/ 217 *cos|--------------|| + \/ 217 *sin|--------------|
        2     \     2                 \      2       //              \      2       /
x3=−32+2174sin⁡(atan⁡(173)2)+i(−332+2174cos⁡(atan⁡(173)2))x_{3} = - \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)
Simplify 
              /    ___              /    /     ___\\\              /    /     ___\\
        3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
x_4 = - - + I*|------- + \/ 217 *cos|--------------|| - \/ 217 *sin|--------------|
        2     \   2                 \      2       //              \      2       /
x4=−2174sin⁡(atan⁡(173)2)−32+i(332+2174cos⁡(atan⁡(173)2))x_{4} = - \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(\frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)
Simplify 
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        3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
x_5 = - - + I*|------- - \/ 217 *cos|--------------|| + \/ 217 *sin|--------------|
        2     \   2                 \      2       //              \      2       /
x5=−32+2174sin⁡(atan⁡(173)2)+i(−2174cos⁡(atan⁡(173)2)+332)x_{5} = - \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + \frac{3 \sqrt{3}}{2}\right)
Simplify 
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        3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
x_6 = - - + I*|- ------- - \/ 217 *cos|--------------|| - \/ 217 *sin|--------------|
        2     \     2                 \      2       //              \      2       /
x6=−2174sin⁡(atan⁡(173)2)−32+i(−2174cos⁡(atan⁡(173)2)−332)x_{6} = - \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3 \sqrt{3}}{2}\right)
Simplify
Sum and product of roots [src] 
sum
                /      ___              /    /     ___\\\              /    /     ___\\           /    ___              /    /     ___\\\              /    /     ___\\           /    ___              /    /     ___\\\              /    /     ___\\           /      ___              /    /     ___\\\              /    /     ___\\
          3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
2 + 4 + - - + I*|- ------- + \/ 217 *cos|--------------|| + \/ 217 *sin|--------------| + - - + I*|------- + \/ 217 *cos|--------------|| - \/ 217 *sin|--------------| + - - + I*|------- - \/ 217 *cos|--------------|| + \/ 217 *sin|--------------| + - - + I*|- ------- - \/ 217 *cos|--------------|| - \/ 217 *sin|--------------|
          2     \     2                 \      2       //              \      2       /     2     \   2                 \      2       //              \      2       /     2     \   2                 \      2       //              \      2       /     2     \     2                 \      2       //              \      2       /
(2)+(4)+(−32+2174sin⁡(atan⁡(173)2)+i(−332+2174cos⁡(atan⁡(173)2)))+(−2174sin⁡(atan⁡(173)2)−32+i(332+2174cos⁡(atan⁡(173)2)))+(−32+2174sin⁡(atan⁡(173)2)+i(−2174cos⁡(atan⁡(173)2)+332))+(−2174sin⁡(atan⁡(173)2)−32+i(−2174cos⁡(atan⁡(173)2)−332))\left(2\right) + \left(4\right) + \left(- \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)\right) + \left(- \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(\frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)\right) + \left(- \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + \frac{3 \sqrt{3}}{2}\right)\right) + \left(- \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3 \sqrt{3}}{2}\right)\right) 
=
  /      ___              /    /     ___\\\     /      ___              /    /     ___\\\     /    ___              /    /     ___\\\     /    ___              /    /     ___\\\
  |  3*\/ 3    4 _____    |atan\17*\/ 3 /||     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||     |3*\/ 3    4 _____    |atan\17*\/ 3 /||     |3*\/ 3    4 _____    |atan\17*\/ 3 /||
I*|- ------- + \/ 217 *cos|--------------|| + I*|- ------- - \/ 217 *cos|--------------|| + I*|------- + \/ 217 *cos|--------------|| + I*|------- - \/ 217 *cos|--------------||
  \     2                 \      2       //     \     2                 \      2       //     \   2                 \      2       //     \   2                 \      2       //
i(−2174cos⁡(atan⁡(173)2)−332)+i(−2174cos⁡(atan⁡(173)2)+332)+i(−332+2174cos⁡(atan⁡(173)2))+i(332+2174cos⁡(atan⁡(173)2))i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3 \sqrt{3}}{2}\right) + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + \frac{3 \sqrt{3}}{2}\right) + i \left(- \frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right) + i \left(\frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)
Simplify 
product
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          3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|     3     |  3*\/ 3    4 _____    |atan\17*\/ 3 /||   4 _____    |atan\17*\/ 3 /|
2 * 4 * - - + I*|- ------- + \/ 217 *cos|--------------|| + \/ 217 *sin|--------------| * - - + I*|------- + \/ 217 *cos|--------------|| - \/ 217 *sin|--------------| * - - + I*|------- - \/ 217 *cos|--------------|| + \/ 217 *sin|--------------| * - - + I*|- ------- - \/ 217 *cos|--------------|| - \/ 217 *sin|--------------|
          2     \     2                 \      2       //              \      2       /     2     \   2                 \      2       //              \      2       /     2     \   2                 \      2       //              \      2       /     2     \     2                 \      2       //              \      2       /
(2)∗(4)∗(−32+2174sin⁡(atan⁡(173)2)+i(−332+2174cos⁡(atan⁡(173)2)))∗(−2174sin⁡(atan⁡(173)2)−32+i(332+2174cos⁡(atan⁡(173)2)))∗(−32+2174sin⁡(atan⁡(173)2)+i(−2174cos⁡(atan⁡(173)2)+332))∗(−2174sin⁡(atan⁡(173)2)−32+i(−2174cos⁡(atan⁡(173)2)−332))\left(2\right) * \left(4\right) * \left(- \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)\right) * \left(- \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(\frac{3 \sqrt{3}}{2} + \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)}\right)\right) * \left(- \frac{3}{2} + \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} + \frac{3 \sqrt{3}}{2}\right)\right) * \left(- \sqrt[4]{217} \sin{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3}{2} + i \left(- \sqrt[4]{217} \cos{\left(\frac{\operatorname{atan}{\left(17 \sqrt{3} \right)}}{2} \right)} - \frac{3 \sqrt{3}}{2}\right)\right) 
=
512
512512
Simplify
Numerical answer [src] 
x1 = 1.16748194582983 + 0.161536067813473*i
x2 = 1.16748194582983 - 0.161536067813473*i
x3 = -4.16748194582983 + 5.3576884905201*i
x4 = -4.16748194582983 - 5.3576884905201*i
x5 = 2.0
x6 = 4.0
x6 = 4.0
The graph
x^6=(6x-8)^3 equation
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