x^4-23x^2+1=

Solution for x^4-23x^2+1= equation:

Simplifying
x4 + -23x2 + 1 = 0

Reorder the terms:
1 + -23x2 + x4 = 0

Solving
1 + -23x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -23x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -23x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -23x2 + x4 = 0 + -1
-23x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-23x2 + x4 = -1

The x term is -23x2.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23x2 + 132.25 + x4 = -1 + 132.25

Reorder the terms:
132.25 + -23x2 + x4 = -1 + 132.25

Combine like terms: -1 + 132.25 = 131.25
132.25 + -23x2 + x4 = 131.25

Factor a perfect square on the left side:
(x2 + -11.5)(x2 + -11.5) = 131.25

Calculate the square root of the right side: 11.456439237

Break this problem into two subproblems by setting 
(x2 + -11.5) equal to 11.456439237 and -11.456439237.

Subproblem 1
x2 + -11.5 = 11.456439237

Simplifying
x2 + -11.5 = 11.456439237

Reorder the terms:
-11.5 + x2 = 11.456439237

Solving
-11.5 + x2 = 11.456439237

Solving for variable 'x'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + x2 = 11.456439237 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + x2 = 11.456439237 + 11.5
x2 = 11.456439237 + 11.5

Combine like terms: 11.456439237 + 11.5 = 22.956439237
x2 = 22.956439237

Simplifying
x2 = 22.956439237

Take the square root of each side:
x = {-4.791287847, 4.791287847}

Subproblem 2
x2 + -11.5 = -11.456439237

Simplifying
x2 + -11.5 = -11.456439237

Reorder the terms:
-11.5 + x2 = -11.456439237

Solving
-11.5 + x2 = -11.456439237

Solving for variable 'x'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + x2 = -11.456439237 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + x2 = -11.456439237 + 11.5
x2 = -11.456439237 + 11.5

Combine like terms: -11.456439237 + 11.5 = 0.043560763
x2 = 0.043560763

Simplifying
x2 = 0.043560763

Take the square root of each side:
x = {-0.208712153, 0.208712153}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4.791287847, 4.791287847, -0.208712153, 0.208712153}