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Simplifying 9n2 + -10n + -5 = 3 Reorder the terms: -5 + -10n + 9n2 = 3 Solving -5 + -10n + 9n2 = 3 Solving for variable 'n'. Reorder the terms: -5 + -3 + -10n + 9n2 = 3 + -3 Combine like terms: -5 + -3 = -8 -8 + -10n + 9n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -8 + -10n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.8888888889 + -1.111111111n + n2 = 0 Move the constant term to the right: Add '0.8888888889' to each side of the equation. -0.8888888889 + -1.111111111n + 0.8888888889 + n2 = 0 + 0.8888888889 Reorder the terms: -0.8888888889 + 0.8888888889 + -1.111111111n + n2 = 0 + 0.8888888889 Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000 0.0000000000 + -1.111111111n + n2 = 0 + 0.8888888889 -1.111111111n + n2 = 0 + 0.8888888889 Combine like terms: 0 + 0.8888888889 = 0.8888888889 -1.111111111n + n2 = 0.8888888889 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 = 0.8888888889 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111n + n2 = 0.8888888889 + 0.3086419752 Combine like terms: 0.8888888889 + 0.3086419752 = 1.1975308641 0.3086419752 + -1.111111111n + n2 = 1.1975308641 Factor a perfect square on the left side: (n + -0.5555555555)(n + -0.5555555555) = 1.1975308641 Calculate the square root of the right side: 1.094317533 Break this problem into two subproblems by setting (n + -0.5555555555) equal to 1.094317533 and -1.094317533.Subproblem 1
n + -0.5555555555 = 1.094317533 Simplifying n + -0.5555555555 = 1.094317533 Reorder the terms: -0.5555555555 + n = 1.094317533 Solving -0.5555555555 + n = 1.094317533 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 = 1.094317533 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = 1.094317533 + 0.5555555555 n = 1.094317533 + 0.5555555555 Combine like terms: 1.094317533 + 0.5555555555 = 1.6498730885 n = 1.6498730885 Simplifying n = 1.6498730885Subproblem 2
n + -0.5555555555 = -1.094317533 Simplifying n + -0.5555555555 = -1.094317533 Reorder the terms: -0.5555555555 + n = -1.094317533 Solving -0.5555555555 + n = -1.094317533 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 = -1.094317533 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = -1.094317533 + 0.5555555555 n = -1.094317533 + 0.5555555555 Combine like terms: -1.094317533 + 0.5555555555 = -0.5387619775 n = -0.5387619775 Simplifying n = -0.5387619775Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.6498730885, -0.5387619775}
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