x^4-3x^2+1=0

Solution for x^4-3x^2+1=0 equation:

Simplifying
x4 + -3x2 + 1 = 0

Reorder the terms:
1 + -3x2 + x4 = 0

Solving
1 + -3x2 + 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 + -3x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -3x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3x2 + x4 = 0 + -1
-3x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-3x2 + x4 = -1

The x term is -3x2.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x2 + 2.25 + x4 = -1 + 2.25

Reorder the terms:
2.25 + -3x2 + x4 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3x2 + x4 = 1.25

Factor a perfect square on the left side:
(x2 + -1.5)(x2 + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x2 + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x2 + -1.5 = 1.118033989

Simplifying
x2 + -1.5 = 1.118033989

Reorder the terms:
-1.5 + x2 = 1.118033989

Solving
-1.5 + x2 = 1.118033989

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x2 = 1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = 1.118033989 + 1.5
x2 = 1.118033989 + 1.5

Combine like terms: 1.118033989 + 1.5 = 2.618033989
x2 = 2.618033989

Simplifying
x2 = 2.618033989

Take the square root of each side:
x = {-1.618033989, 1.618033989}

Subproblem 2
x2 + -1.5 = -1.118033989

Simplifying
x2 + -1.5 = -1.118033989

Reorder the terms:
-1.5 + x2 = -1.118033989

Solving
-1.5 + x2 = -1.118033989

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x2 = -1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x2 = -1.118033989 + 1.5
x2 = -1.118033989 + 1.5

Combine like terms: -1.118033989 + 1.5 = 0.381966011
x2 = 0.381966011

Simplifying
x2 = 0.381966011

Take the square root of each side:
x = {-0.618033989, 0.618033989}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.618033989, 1.618033989, -0.618033989, 0.618033989}