5x^2-20x^4+21x^2-2=0

Solution for 5x^2-20x^4+21x^2-2=0 equation:

Simplifying
5x2 + -20x4 + 21x2 + -2 = 0

Reorder the terms:
-2 + 5x2 + 21x2 + -20x4 = 0

Combine like terms: 5x2 + 21x2 = 26x2
-2 + 26x2 + -20x4 = 0

Solving
-2 + 26x2 + -20x4 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 13x2 + -10x4) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1 + 13x2 + -10x4)' equal to zero and attempt to solve:

Simplifying
-1 + 13x2 + -10x4 = 0

Solving
-1 + 13x2 + -10x4 = 0

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
0.1 + -1.3x2 + x4 = 0.0

Move the constant term to the right:

Add '-0.1' to each side of the equation.
0.1 + -1.3x2 + -0.1 + x4 = 0.0 + -0.1

Reorder the terms:
0.1 + -0.1 + -1.3x2 + x4 = 0.0 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + -1.3x2 + x4 = 0.0 + -0.1
-1.3x2 + x4 = 0.0 + -0.1

Combine like terms: 0.0 + -0.1 = -0.1
-1.3x2 + x4 = -0.1

The x term is -1.3x2.  Take half its coefficient (-0.65).
Square it (0.4225) and add it to both sides.

Add '0.4225' to each side of the equation.
-1.3x2 + 0.4225 + x4 = -0.1 + 0.4225

Reorder the terms:
0.4225 + -1.3x2 + x4 = -0.1 + 0.4225

Combine like terms: -0.1 + 0.4225 = 0.3225
0.4225 + -1.3x2 + x4 = 0.3225

Factor a perfect square on the left side:
(x2 + -0.65)(x2 + -0.65) = 0.3225

Calculate the square root of the right side: 0.567890835

Break this problem into two subproblems by setting 
(x2 + -0.65) equal to 0.567890835 and -0.567890835.

Subproblem 1
x2 + -0.65 = 0.567890835

Simplifying
x2 + -0.65 = 0.567890835

Reorder the terms:
-0.65 + x2 = 0.567890835

Solving
-0.65 + x2 = 0.567890835

Solving for variable 'x'.

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

Add '0.65' to each side of the equation.
-0.65 + 0.65 + x2 = 0.567890835 + 0.65

Combine like terms: -0.65 + 0.65 = 0.00
0.00 + x2 = 0.567890835 + 0.65
x2 = 0.567890835 + 0.65

Combine like terms: 0.567890835 + 0.65 = 1.217890835
x2 = 1.217890835

Simplifying
x2 = 1.217890835

Take the square root of each side:
x = {-1.103580915, 1.103580915}

Subproblem 2
x2 + -0.65 = -0.567890835

Simplifying
x2 + -0.65 = -0.567890835

Reorder the terms:
-0.65 + x2 = -0.567890835

Solving
-0.65 + x2 = -0.567890835

Solving for variable 'x'.

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

Add '0.65' to each side of the equation.
-0.65 + 0.65 + x2 = -0.567890835 + 0.65

Combine like terms: -0.65 + 0.65 = 0.00
0.00 + x2 = -0.567890835 + 0.65
x2 = -0.567890835 + 0.65

Combine like terms: -0.567890835 + 0.65 = 0.082109165
x2 = 0.082109165

Simplifying
x2 = 0.082109165

Take the square root of each side:
x = {-0.286546968, 0.286546968}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.103580915, 1.103580915, -0.286546968, 0.286546968}
Solution
x = {-1.103580915, 1.103580915, -0.286546968, 0.286546968}