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Simplifying 2c(c + -1) = 1 Reorder the terms: 2c(-1 + c) = 1 (-1 * 2c + c * 2c) = 1 (-2c + 2c2) = 1 Solving -2c + 2c2 = 1 Solving for variable 'c'. Reorder the terms: -1 + -2c + 2c2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -2c + 2c2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -1c + c2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1c + 0.5 + c2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1c + c2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1c + c2 = 0 + 0.5 -1c + c2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1c + c2 = 0.5 The c term is -1c. 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. -1c + 0.25 + c2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1c + c2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1c + c2 = 0.75 Factor a perfect square on the left side: (c + -0.5)(c + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (c + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
c + -0.5 = 0.866025404 Simplifying c + -0.5 = 0.866025404 Reorder the terms: -0.5 + c = 0.866025404 Solving -0.5 + c = 0.866025404 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + c = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + c = 0.866025404 + 0.5 c = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 c = 1.366025404 Simplifying c = 1.366025404Subproblem 2
c + -0.5 = -0.866025404 Simplifying c + -0.5 = -0.866025404 Reorder the terms: -0.5 + c = -0.866025404 Solving -0.5 + c = -0.866025404 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + c = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + c = -0.866025404 + 0.5 c = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 c = -0.366025404 Simplifying c = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.366025404, -0.366025404}
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