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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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