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Simplifying 2u2 + -2u + -1 = 0 Reorder the terms: -1 + -2u + 2u2 = 0 Solving -1 + -2u + 2u2 = 0 Solving for variable 'u'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -1u + u2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1u + 0.5 + u2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1u + u2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1u + u2 = 0 + 0.5 -1u + u2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1u + u2 = 0.5 The u term is -1u. 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. -1u + 0.25 + u2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1u + u2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1u + u2 = 0.75 Factor a perfect square on the left side: (u + -0.5)(u + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (u + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
u + -0.5 = 0.866025404 Simplifying u + -0.5 = 0.866025404 Reorder the terms: -0.5 + u = 0.866025404 Solving -0.5 + u = 0.866025404 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + u = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + u = 0.866025404 + 0.5 u = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 u = 1.366025404 Simplifying u = 1.366025404Subproblem 2
u + -0.5 = -0.866025404 Simplifying u + -0.5 = -0.866025404 Reorder the terms: -0.5 + u = -0.866025404 Solving -0.5 + u = -0.866025404 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + u = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + u = -0.866025404 + 0.5 u = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 u = -0.366025404 Simplifying u = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. u = {1.366025404, -0.366025404}
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