.25x^4-x^2+1=0

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Solution for .25x^4-x^2+1=0 equation:


Simplifying
0.25x4 + -1x2 + 1 = 0

Reorder the terms:
1 + -1x2 + 0.25x4 = 0

Solving
1 + -1x2 + 0.25x4 = 0

Solving for variable 'x'.

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

Divide each side by '0.25'.
4 + -4x2 + x4 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -4x2 + -4 + x4 = 0 + -4

Reorder the terms:
4 + -4 + -4x2 + x4 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -4x2 + x4 = 0 + -4
-4x2 + x4 = 0 + -4

Combine like terms: 0 + -4 = -4
-4x2 + x4 = -4

The x term is -4x2.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x2 + 4 + x4 = -4 + 4

Reorder the terms:
4 + -4x2 + x4 = -4 + 4

Combine like terms: -4 + 4 = 0
4 + -4x2 + x4 = 0

Factor a perfect square on the left side:
(x2 + -2)(x2 + -2) = 0

Calculate the square root of the right side: 0

Break this problem into two subproblems by setting 
(x2 + -2) equal to 0 and 0.

Subproblem 1

x2 + -2 = 0 Simplifying x2 + -2 = 0 Reorder the terms: -2 + x2 = 0 Solving -2 + x2 = 0 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 0 + 2 x2 = 0 + 2 Combine like terms: 0 + 2 = 2 x2 = 2 Simplifying x2 = 2 Take the square root of each side: x = {-1.414213562, 1.414213562}

Subproblem 2

x2 + -2 = 0 Simplifying x2 + -2 = 0 Reorder the terms: -2 + x2 = 0 Solving -2 + x2 = 0 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + x2 = 0 + 2 x2 = 0 + 2 Combine like terms: 0 + 2 = 2 x2 = 2 Simplifying x2 = 2 Take the square root of each side: x = {-1.414213562, 1.414213562}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.414213562, 1.414213562, -1.414213562, 1.414213562}

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