2X^4-10X^2+11=0

Solution for 2X^4-10X^2+11=0 equation:

Simplifying
2X4 + -10X2 + 11 = 0

Reorder the terms:
11 + -10X2 + 2X4 = 0

Solving
11 + -10X2 + 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'.
5.5 + -5X2 + X4 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + -5X2 + -5.5 + X4 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + -5X2 + X4 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + -5X2 + X4 = 0 + -5.5
-5X2 + X4 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
-5X2 + X4 = -5.5

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 = -5.5 + 6.25

Reorder the terms:
6.25 + -5X2 + X4 = -5.5 + 6.25

Combine like terms: -5.5 + 6.25 = 0.75
6.25 + -5X2 + X4 = 0.75

Factor a perfect square on the left side:
(X2 + -2.5)(X2 + -2.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(X2 + -2.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
X2 + -2.5 = 0.866025404

Simplifying
X2 + -2.5 = 0.866025404

Reorder the terms:
-2.5 + X2 = 0.866025404

Solving
-2.5 + X2 = 0.866025404

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 = 0.866025404 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + X2 = 0.866025404 + 2.5
X2 = 0.866025404 + 2.5

Combine like terms: 0.866025404 + 2.5 = 3.366025404
X2 = 3.366025404

Simplifying
X2 = 3.366025404

Take the square root of each side:
X = {-1.834673105, 1.834673105}

Subproblem 2
X2 + -2.5 = -0.866025404

Simplifying
X2 + -2.5 = -0.866025404

Reorder the terms:
-2.5 + X2 = -0.866025404

Solving
-2.5 + X2 = -0.866025404

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 = -0.866025404 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + X2 = -0.866025404 + 2.5
X2 = -0.866025404 + 2.5

Combine like terms: -0.866025404 + 2.5 = 1.633974596
X2 = 1.633974596

Simplifying
X2 = 1.633974596

Take the square root of each side:
X = {-1.278270158, 1.278270158}

Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {-1.834673105, 1.834673105, -1.278270158, 1.278270158}