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Simplifying 4x4 + -8x2 + 1 = 0 Reorder the terms: 1 + -8x2 + 4x4 = 0 Solving 1 + -8x2 + 4x4 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -2x2 + x4 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -2x2 + -0.25 + x4 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -2x2 + x4 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -2x2 + x4 = 0 + -0.25 -2x2 + x4 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -2x2 + x4 = -0.25 The x term is -2x2. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x2 + 1 + x4 = -0.25 + 1 Reorder the terms: 1 + -2x2 + x4 = -0.25 + 1 Combine like terms: -0.25 + 1 = 0.75 1 + -2x2 + x4 = 0.75 Factor a perfect square on the left side: (x2 + -1)(x2 + -1) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x2 + -1) equal to 0.866025404 and -0.866025404.Subproblem 1
x2 + -1 = 0.866025404 Simplifying x2 + -1 = 0.866025404 Reorder the terms: -1 + x2 = 0.866025404 Solving -1 + x2 = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = 0.866025404 + 1 x2 = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 x2 = 1.866025404 Simplifying x2 = 1.866025404 Take the square root of each side: x = {-1.366025404, 1.366025404}Subproblem 2
x2 + -1 = -0.866025404 Simplifying x2 + -1 = -0.866025404 Reorder the terms: -1 + x2 = -0.866025404 Solving -1 + x2 = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = -0.866025404 + 1 x2 = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 x2 = 0.133974596 Simplifying x2 = 0.133974596 Take the square root of each side: x = {-0.366025403, 0.366025403}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.366025404, 1.366025404, -0.366025403, 0.366025403}
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