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Simplifying 2n2 + 10n + 11 = 0 Reorder the terms: 11 + 10n + 2n2 = 0 Solving 11 + 10n + 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'. 5.5 + 5n + n2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + 5n + -5.5 + n2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + 5n + n2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + 5n + n2 = 0 + -5.5 5n + n2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 5n + n2 = -5.5 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = -5.5 + 6.25 Reorder the terms: 6.25 + 5n + n2 = -5.5 + 6.25 Combine like terms: -5.5 + 6.25 = 0.75 6.25 + 5n + n2 = 0.75 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (n + 2.5) equal to 0.866025404 and -0.866025404.Subproblem 1
n + 2.5 = 0.866025404 Simplifying n + 2.5 = 0.866025404 Reorder the terms: 2.5 + n = 0.866025404 Solving 2.5 + n = 0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 0.866025404 + -2.5 n = 0.866025404 + -2.5 Combine like terms: 0.866025404 + -2.5 = -1.633974596 n = -1.633974596 Simplifying n = -1.633974596Subproblem 2
n + 2.5 = -0.866025404 Simplifying n + 2.5 = -0.866025404 Reorder the terms: 2.5 + n = -0.866025404 Solving 2.5 + n = -0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -0.866025404 + -2.5 n = -0.866025404 + -2.5 Combine like terms: -0.866025404 + -2.5 = -3.366025404 n = -3.366025404 Simplifying n = -3.366025404Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.633974596, -3.366025404}
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