4x^4+1=20x^2

Solution for 4x^4+1=20x^2 equation:

Simplifying
4x4 + 1 = 20x2

Reorder the terms:
1 + 4x4 = 20x2

Solving
1 + 4x4 = 20x2

Solving for variable 'x'.

Reorder the terms:
1 + -20x2 + 4x4 = 20x2 + -20x2

Combine like terms: 20x2 + -20x2 = 0
1 + -20x2 + 4x4 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
0.25 + -5x2 + x4 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -5x2 + -0.25 + x4 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -5x2 + x4 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -5x2 + x4 = 0 + -0.25
-5x2 + x4 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-5x2 + x4 = -0.25

The x term is -5x2.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x2 + 6.25 + x4 = -0.25 + 6.25

Reorder the terms:
6.25 + -5x2 + x4 = -0.25 + 6.25

Combine like terms: -0.25 + 6.25 = 6
6.25 + -5x2 + x4 = 6

Factor a perfect square on the left side:
(x2 + -2.5)(x2 + -2.5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x2 + -2.5) equal to 2.449489743 and -2.449489743.

Subproblem 1
x2 + -2.5 = 2.449489743

Simplifying
x2 + -2.5 = 2.449489743

Reorder the terms:
-2.5 + x2 = 2.449489743

Solving
-2.5 + x2 = 2.449489743

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x2 = 2.449489743 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x2 = 2.449489743 + 2.5
x2 = 2.449489743 + 2.5

Combine like terms: 2.449489743 + 2.5 = 4.949489743
x2 = 4.949489743

Simplifying
x2 = 4.949489743

Take the square root of each side:
x = {-2.224744871, 2.224744871}

Subproblem 2
x2 + -2.5 = -2.449489743

Simplifying
x2 + -2.5 = -2.449489743

Reorder the terms:
-2.5 + x2 = -2.449489743

Solving
-2.5 + x2 = -2.449489743

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x2 = -2.449489743 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x2 = -2.449489743 + 2.5
x2 = -2.449489743 + 2.5

Combine like terms: -2.449489743 + 2.5 = 0.050510257
x2 = 0.050510257

Simplifying
x2 = 0.050510257

Take the square root of each side:
x = {-0.224744871, 0.224744871}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.224744871, 2.224744871, -0.224744871, 0.224744871}