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

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


Simplifying
0 = x4 + -3x2 + 1

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

Solving
0 = 1 + -3x2 + x4

Solving for variable 'x'.

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

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

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

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

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -3x2 + x4 = 0

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}

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