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