x^4-41x^2-20=0

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Solution for x^4-41x^2-20=0 equation: 


Simplifying
x4 + -41x2 + -20 = 0

Reorder the terms:
-20 + -41x2 + x4 = 0

Solving
-20 + -41x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -41x2 + 20 + x4 = 0 + 20

Reorder the terms:
-20 + 20 + -41x2 + x4 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -41x2 + x4 = 0 + 20
-41x2 + x4 = 0 + 20

Combine like terms: 0 + 20 = 20
-41x2 + x4 = 20

The x term is -41x2.  Take half its coefficient (-20.5).
Square it (420.25) and add it to both sides.

Add '420.25' to each side of the equation.
-41x2 + 420.25 + x4 = 20 + 420.25

Reorder the terms:
420.25 + -41x2 + x4 = 20 + 420.25

Combine like terms: 20 + 420.25 = 440.25
420.25 + -41x2 + x4 = 440.25

Factor a perfect square on the left side:
(x2 + -20.5)(x2 + -20.5) = 440.25

Calculate the square root of the right side: 20.982135258

Break this problem into two subproblems by setting 
(x2 + -20.5) equal to 20.982135258 and -20.982135258.


Subproblem 1
x2 + -20.5 = 20.982135258

Simplifying
x2 + -20.5 = 20.982135258

Reorder the terms:
-20.5 + x2 = 20.982135258

Solving
-20.5 + x2 = 20.982135258

Solving for variable 'x'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + x2 = 20.982135258 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + x2 = 20.982135258 + 20.5
x2 = 20.982135258 + 20.5

Combine like terms: 20.982135258 + 20.5 = 41.482135258
x2 = 41.482135258

Simplifying
x2 = 41.482135258

Take the square root of each side:
x = {-6.440662641, 6.440662641}


Subproblem 2
x2 + -20.5 = -20.982135258

Simplifying
x2 + -20.5 = -20.982135258

Reorder the terms:
-20.5 + x2 = -20.982135258

Solving
-20.5 + x2 = -20.982135258

Solving for variable 'x'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + x2 = -20.982135258 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + x2 = -20.982135258 + 20.5
x2 = -20.982135258 + 20.5

Combine like terms: -20.982135258 + 20.5 = -0.482135258
x2 = -0.482135258

Simplifying
x2 = -0.482135258

Reorder the terms:
0.482135258 + x2 = -0.482135258 + 0.482135258

Combine like terms: -0.482135258 + 0.482135258 = 0.000000000
0.482135258 + 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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