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Simplifying 9n2 = -10n + 9 Reorder the terms: 9n2 = 9 + -10n Solving 9n2 = 9 + -10n Solving for variable 'n'. Reorder the terms: -9 + 10n + 9n2 = 9 + -10n + -9 + 10n Reorder the terms: -9 + 10n + 9n2 = 9 + -9 + -10n + 10n Combine like terms: 9 + -9 = 0 -9 + 10n + 9n2 = 0 + -10n + 10n -9 + 10n + 9n2 = -10n + 10n Combine like terms: -10n + 10n = 0 -9 + 10n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1 + 1.111111111n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.111111111n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 1.111111111n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.111111111n + n2 = 0 + 1 1.111111111n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.111111111n + n2 = 1 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 + 0.3086419752 Reorder the terms: 0.3086419752 + 1.111111111n + n2 = 1 + 0.3086419752 Combine like terms: 1 + 0.3086419752 = 1.3086419752 0.3086419752 + 1.111111111n + n2 = 1.3086419752 Factor a perfect square on the left side: (n + 0.5555555555)(n + 0.5555555555) = 1.3086419752 Calculate the square root of the right side: 1.143958905 Break this problem into two subproblems by setting (n + 0.5555555555) equal to 1.143958905 and -1.143958905.Subproblem 1
n + 0.5555555555 = 1.143958905 Simplifying n + 0.5555555555 = 1.143958905 Reorder the terms: 0.5555555555 + n = 1.143958905 Solving 0.5555555555 + n = 1.143958905 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 = 1.143958905 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = 1.143958905 + -0.5555555555 n = 1.143958905 + -0.5555555555 Combine like terms: 1.143958905 + -0.5555555555 = 0.5884033495 n = 0.5884033495 Simplifying n = 0.5884033495Subproblem 2
n + 0.5555555555 = -1.143958905 Simplifying n + 0.5555555555 = -1.143958905 Reorder the terms: 0.5555555555 + n = -1.143958905 Solving 0.5555555555 + n = -1.143958905 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 = -1.143958905 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + n = -1.143958905 + -0.5555555555 n = -1.143958905 + -0.5555555555 Combine like terms: -1.143958905 + -0.5555555555 = -1.6995144605 n = -1.6995144605 Simplifying n = -1.6995144605Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.5884033495, -1.6995144605}
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