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Simplifying -9n2 + -10n = -11 Reorder the terms: -10n + -9n2 = -11 Solving -10n + -9n2 = -11 Solving for variable 'n'. Reorder the terms: 11 + -10n + -9n2 = -11 + 11 Combine like terms: -11 + 11 = 0 11 + -10n + -9n2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -1.222222222 + 1.111111111n + n2 = 0 Move the constant term to the right: Add '1.222222222' to each side of the equation. -1.222222222 + 1.111111111n + 1.222222222 + n2 = 0 + 1.222222222 Reorder the terms: -1.222222222 + 1.222222222 + 1.111111111n + n2 = 0 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + 1.111111111n + n2 = 0 + 1.222222222 1.111111111n + n2 = 0 + 1.222222222 Combine like terms: 0 + 1.222222222 = 1.222222222 1.111111111n + n2 = 1.222222222 The n term is 1.111111111n. Take half its coefficient (0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. 1.111111111n + 0.3086419752 + n2 = 1.222222222 + 0.3086419752 Reorder the terms: 0.3086419752 + 1.111111111n + n2 = 1.222222222 + 0.3086419752 Combine like terms: 1.222222222 + 0.3086419752 = 1.5308641972 0.3086419752 + 1.111111111n + n2 = 1.5308641972 Factor a perfect square on the left side: (n + 0.5555555555)(n + 0.5555555555) = 1.5308641972 Calculate the square root of the right side: 1.237280969 Break this problem into two subproblems by setting (n + 0.5555555555) equal to 1.237280969 and -1.237280969.Subproblem 1
n + 0.5555555555 = 1.237280969 Simplifying n + 0.5555555555 = 1.237280969 Reorder the terms: 0.5555555555 + n = 1.237280969 Solving 0.5555555555 + n = 1.237280969 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + n = 1.237280969 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = 1.237280969 + -0.5555555555 n = 1.237280969 + -0.5555555555 Combine like terms: 1.237280969 + -0.5555555555 = 0.6817254135 n = 0.6817254135 Simplifying n = 0.6817254135Subproblem 2
n + 0.5555555555 = -1.237280969 Simplifying n + 0.5555555555 = -1.237280969 Reorder the terms: 0.5555555555 + n = -1.237280969 Solving 0.5555555555 + n = -1.237280969 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + n = -1.237280969 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = -1.237280969 + -0.5555555555 n = -1.237280969 + -0.5555555555 Combine like terms: -1.237280969 + -0.5555555555 = -1.7928365245 n = -1.7928365245 Simplifying n = -1.7928365245Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.6817254135, -1.7928365245}
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