x^6+1=52x^3

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Solution for x^6+1=52x^3 equation: 


Simplifying
x6 + 1 = 52x3

Reorder the terms:
1 + x6 = 52x3

Solving
1 + x6 = 52x3

Solving for variable 'x'.

Reorder the terms:
1 + -52x3 + x6 = 52x3 + -52x3

Combine like terms: 52x3 + -52x3 = 0
1 + -52x3 + x6 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -52x3 + -1 + x6 = 0 + -1

Reorder the terms:
1 + -1 + -52x3 + x6 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -52x3 + x6 = 0 + -1
-52x3 + x6 = 0 + -1

Combine like terms: 0 + -1 = -1
-52x3 + x6 = -1

The x term is -52x3.  Take half its coefficient (-26).
Square it (676) and add it to both sides.

Add '676' to each side of the equation.
-52x3 + 676 + x6 = -1 + 676

Reorder the terms:
676 + -52x3 + x6 = -1 + 676

Combine like terms: -1 + 676 = 675
676 + -52x3 + x6 = 675

Factor a perfect square on the left side:
(x3 + -26)(x3 + -26) = 675

Calculate the square root of the right side: 25.980762114

Break this problem into two subproblems by setting 
(x3 + -26) equal to 25.980762114 and -25.980762114.


Subproblem 1
x3 + -26 = 25.980762114

Simplifying
x3 + -26 = 25.980762114

Reorder the terms:
-26 + x3 = 25.980762114

Solving
-26 + x3 = 25.980762114

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '26' to each side of the equation.
-26 + 26 + x3 = 25.980762114 + 26

Combine like terms: -26 + 26 = 0
0 + x3 = 25.980762114 + 26
x3 = 25.980762114 + 26

Combine like terms: 25.980762114 + 26 = 51.980762114
x3 = 51.980762114

Simplifying
x3 = 51.980762114

Reorder the terms:
-51.980762114 + x3 = 51.980762114 + -51.980762114

Combine like terms: 51.980762114 + -51.980762114 = 0.000000000
-51.980762114 + x3 = 0.000000000

The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.


Subproblem 2
x3 + -26 = -25.980762114

Simplifying
x3 + -26 = -25.980762114

Reorder the terms:
-26 + x3 = -25.980762114

Solving
-26 + x3 = -25.980762114

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '26' to each side of the equation.
-26 + 26 + x3 = -25.980762114 + 26

Combine like terms: -26 + 26 = 0
0 + x3 = -25.980762114 + 26
x3 = -25.980762114 + 26

Combine like terms: -25.980762114 + 26 = 0.019237886
x3 = 0.019237886

Simplifying
x3 = 0.019237886

Reorder the terms:
-0.019237886 + x3 = 0.019237886 + -0.019237886

Combine like terms: 0.019237886 + -0.019237886 = 0.000000000
-0.019237886 + x3 = 0.000000000

The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.

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