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Simplifying 2x(x + 1) = 1 Reorder the terms: 2x(1 + x) = 1 (1 * 2x + x * 2x) = 1 (2x + 2x2) = 1 Solving 2x + 2x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 2x + 2x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x + x2 = 0 + 0.5 x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 x + x2 = 0.5 The x term is x. 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. + 0.25 + x2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + x2 = 0.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.5 = 0.866025404 Simplifying x + 0.5 = 0.866025404 Reorder the terms: 0.5 + x = 0.866025404 Solving 0.5 + x = 0.866025404 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 + x = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.866025404 + -0.5 x = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 x = 0.366025404 Simplifying x = 0.366025404Subproblem 2
x + 0.5 = -0.866025404 Simplifying x + 0.5 = -0.866025404 Reorder the terms: 0.5 + x = -0.866025404 Solving 0.5 + x = -0.866025404 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 + x = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.866025404 + -0.5 x = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 x = -1.366025404 Simplifying x = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.366025404, -1.366025404}
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