X^4-27X^2-4=0

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Solution for X^4-27X^2-4=0 equation: 


Simplifying
X4 + -27X2 + -4 = 0

Reorder the terms:
-4 + -27X2 + X4 = 0

Solving
-4 + -27X2 + X4 = 0

Solving for variable 'X'.

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -27X2 + 4 + X4 = 0 + 4

Reorder the terms:
-4 + 4 + -27X2 + X4 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -27X2 + X4 = 0 + 4
-27X2 + X4 = 0 + 4

Combine like terms: 0 + 4 = 4
-27X2 + X4 = 4

The X term is -27X2.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27X2 + 182.25 + X4 = 4 + 182.25

Reorder the terms:
182.25 + -27X2 + X4 = 4 + 182.25

Combine like terms: 4 + 182.25 = 186.25
182.25 + -27X2 + X4 = 186.25

Factor a perfect square on the left side:
(X2 + -13.5)(X2 + -13.5) = 186.25

Calculate the square root of the right side: 13.647344064

Break this problem into two subproblems by setting 
(X2 + -13.5) equal to 13.647344064 and -13.647344064.


Subproblem 1
X2 + -13.5 = 13.647344064

Simplifying
X2 + -13.5 = 13.647344064

Reorder the terms:
-13.5 + X2 = 13.647344064

Solving
-13.5 + X2 = 13.647344064

Solving for variable 'X'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + X2 = 13.647344064 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + X2 = 13.647344064 + 13.5
X2 = 13.647344064 + 13.5

Combine like terms: 13.647344064 + 13.5 = 27.147344064
X2 = 27.147344064

Simplifying
X2 = 27.147344064

Take the square root of each side:
X = {-5.210311321, 5.210311321}


Subproblem 2
X2 + -13.5 = -13.647344064

Simplifying
X2 + -13.5 = -13.647344064

Reorder the terms:
-13.5 + X2 = -13.647344064

Solving
-13.5 + X2 = -13.647344064

Solving for variable 'X'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + X2 = -13.647344064 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + X2 = -13.647344064 + 13.5
X2 = -13.647344064 + 13.5

Combine like terms: -13.647344064 + 13.5 = -0.147344064
X2 = -0.147344064

Simplifying
X2 = -0.147344064

Reorder the terms:
0.147344064 + X2 = -0.147344064 + 0.147344064

Combine like terms: -0.147344064 + 0.147344064 = 0.000000000
0.147344064 + X2 = 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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