x^4-x^2+0.25=0

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Solution for x^4-x^2+0.25=0 equation:


Simplifying
x4 + -1x2 + 0.25 = 0

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

Solving
0.25 + -1x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -1x2 + -0.25 + x4 = 0 + -0.25

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

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -1x2 + x4 = 0 + -0.25
-1x2 + x4 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-1x2 + x4 = -0.25

The x term is -1x2.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x2 + 0.25 + x4 = -0.25 + 0.25

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

Combine like terms: -0.25 + 0.25 = 0.00
0.25 + -1x2 + x4 = 0.00

Factor a perfect square on the left side:
(x2 + -0.5)(x2 + -0.5) = 0.00

Calculate the square root of the right side: 0.00

Break this problem into two subproblems by setting 
(x2 + -0.5) equal to 0.00 and 0.00.

Subproblem 1

x2 + -0.5 = 0.00 Simplifying x2 + -0.5 = 0.00 Reorder the terms: -0.5 + x2 = 0.00 Solving -0.5 + x2 = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.00 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.00 + 0.5 x2 = 0.00 + 0.5 Combine like terms: 0.00 + 0.5 = 0.5 x2 = 0.5 Simplifying x2 = 0.5 Take the square root of each side: x = {-0.707106781, 0.707106781}

Subproblem 2

x2 + -0.5 = 0.00 Simplifying x2 + -0.5 = 0.00 Reorder the terms: -0.5 + x2 = 0.00 Solving -0.5 + x2 = 0.00 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x2 = 0.00 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.00 + 0.5 x2 = 0.00 + 0.5 Combine like terms: 0.00 + 0.5 = 0.5 x2 = 0.5 Simplifying x2 = 0.5 Take the square root of each side: x = {-0.707106781, 0.707106781}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.707106781, 0.707106781, -0.707106781, 0.707106781}

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