2x^4-5x^2+1=0

Solution for 2x^4-5x^2+1=0 equation:

Simplifying
2x4 + -5x2 + 1 = 0

Reorder the terms:
1 + -5x2 + 2x4 = 0

Solving
1 + -5x2 + 2x4 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
0.5 + -2.5x2 + x4 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.5x2 + -0.5 + x4 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.5x2 + x4 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.5x2 + x4 = 0 + -0.5
-2.5x2 + x4 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.5x2 + x4 = -0.5

The x term is -2.5x2.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x2 + 1.5625 + x4 = -0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5x2 + x4 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + -2.5x2 + x4 = 1.0625

Factor a perfect square on the left side:
(x2 + -1.25)(x2 + -1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x2 + -1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
x2 + -1.25 = 1.030776406

Simplifying
x2 + -1.25 = 1.030776406

Reorder the terms:
-1.25 + x2 = 1.030776406

Solving
-1.25 + x2 = 1.030776406

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x2 = 1.030776406 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x2 = 1.030776406 + 1.25
x2 = 1.030776406 + 1.25

Combine like terms: 1.030776406 + 1.25 = 2.280776406
x2 = 2.280776406

Simplifying
x2 = 2.280776406

Take the square root of each side:
x = {-1.510223959, 1.510223959}

Subproblem 2
x2 + -1.25 = -1.030776406

Simplifying
x2 + -1.25 = -1.030776406

Reorder the terms:
-1.25 + x2 = -1.030776406

Solving
-1.25 + x2 = -1.030776406

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x2 = -1.030776406 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x2 = -1.030776406 + 1.25
x2 = -1.030776406 + 1.25

Combine like terms: -1.030776406 + 1.25 = 0.219223594
x2 = 0.219223594

Simplifying
x2 = 0.219223594

Take the square root of each side:
x = {-0.468213193, 0.468213193}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.510223959, 1.510223959, -0.468213193, 0.468213193}