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Simplifying 2n2 + 2n + -1 = 0 Reorder the terms: -1 + 2n + 2n2 = 0 Solving -1 + 2n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n + n2 = 0 + 0.5 n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 n + n2 = 0.5 The n term is n. 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 + n2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + n2 = 0.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (n + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
n + 0.5 = 0.866025404 Simplifying n + 0.5 = 0.866025404 Reorder the terms: 0.5 + n = 0.866025404 Solving 0.5 + n = 0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.866025404 + -0.5 n = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 n = 0.366025404 Simplifying n = 0.366025404Subproblem 2
n + 0.5 = -0.866025404 Simplifying n + 0.5 = -0.866025404 Reorder the terms: 0.5 + n = -0.866025404 Solving 0.5 + n = -0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.866025404 + -0.5 n = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 n = -1.366025404 Simplifying n = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.366025404, -1.366025404}
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