8x^4-8x^2+1=0.5

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Solution for 8x^4-8x^2+1=0.5 equation:


Simplifying
8x4 + -8x2 + 1 = 0.5

Reorder the terms:
1 + -8x2 + 8x4 = 0.5

Solving
1 + -8x2 + 8x4 = 0.5

Solving for variable 'x'.

Reorder the terms:
1 + -0.5 + -8x2 + 8x4 = 0.5 + -0.5

Combine like terms: 1 + -0.5 = 0.5
0.5 + -8x2 + 8x4 = 0.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.5 + -8x2 + 8x4 = 0.0

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

Divide each side by '8'.
0.0625 + -1x2 + x4 = 0.0

Move the constant term to the right:

Add '-0.0625' to each side of the equation.
0.0625 + -1x2 + -0.0625 + x4 = 0.0 + -0.0625

Reorder the terms:
0.0625 + -0.0625 + -1x2 + x4 = 0.0 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + -1x2 + x4 = 0.0 + -0.0625
-1x2 + x4 = 0.0 + -0.0625

Combine like terms: 0.0 + -0.0625 = -0.0625
-1x2 + x4 = -0.0625

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 = -0.0625 + 0.25

Reorder the terms:
0.25 + -1x2 + x4 = -0.0625 + 0.25

Combine like terms: -0.0625 + 0.25 = 0.1875
0.25 + -1x2 + x4 = 0.1875

Factor a perfect square on the left side:
(x2 + -0.5)(x2 + -0.5) = 0.1875

Calculate the square root of the right side: 0.433012702

Break this problem into two subproblems by setting 
(x2 + -0.5) equal to 0.433012702 and -0.433012702.

Subproblem 1

x2 + -0.5 = 0.433012702 Simplifying x2 + -0.5 = 0.433012702 Reorder the terms: -0.5 + x2 = 0.433012702 Solving -0.5 + x2 = 0.433012702 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 = 0.433012702 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = 0.433012702 + 0.5 x2 = 0.433012702 + 0.5 Combine like terms: 0.433012702 + 0.5 = 0.933012702 x2 = 0.933012702 Simplifying x2 = 0.933012702 Take the square root of each side: x = {-0.965925826, 0.965925826}

Subproblem 2

x2 + -0.5 = -0.433012702 Simplifying x2 + -0.5 = -0.433012702 Reorder the terms: -0.5 + x2 = -0.433012702 Solving -0.5 + x2 = -0.433012702 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 = -0.433012702 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x2 = -0.433012702 + 0.5 x2 = -0.433012702 + 0.5 Combine like terms: -0.433012702 + 0.5 = 0.066987298 x2 = 0.066987298 Simplifying x2 = 0.066987298 Take the square root of each side: x = {-0.258819045, 0.258819045}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.965925826, 0.965925826, -0.258819045, 0.258819045}

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