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Simplifying 9n2 + -10n = 20 Reorder the terms: -10n + 9n2 = 20 Solving -10n + 9n2 = 20 Solving for variable 'n'. Reorder the terms: -20 + -10n + 9n2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + -10n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.222222222 + -1.111111111n + n2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + -1.111111111n + 2.222222222 + n2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + -1.111111111n + n2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + -1.111111111n + n2 = 0 + 2.222222222 -1.111111111n + n2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 -1.111111111n + n2 = 2.222222222 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 = 2.222222222 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111n + n2 = 2.222222222 + 0.3086419752 Combine like terms: 2.222222222 + 0.3086419752 = 2.5308641972 0.3086419752 + -1.111111111n + n2 = 2.5308641972 Factor a perfect square on the left side: (n + -0.5555555555)(n + -0.5555555555) = 2.5308641972 Calculate the square root of the right side: 1.590869007 Break this problem into two subproblems by setting (n + -0.5555555555) equal to 1.590869007 and -1.590869007.Subproblem 1
n + -0.5555555555 = 1.590869007 Simplifying n + -0.5555555555 = 1.590869007 Reorder the terms: -0.5555555555 + n = 1.590869007 Solving -0.5555555555 + n = 1.590869007 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 = 1.590869007 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = 1.590869007 + 0.5555555555 n = 1.590869007 + 0.5555555555 Combine like terms: 1.590869007 + 0.5555555555 = 2.1464245625 n = 2.1464245625 Simplifying n = 2.1464245625Subproblem 2
n + -0.5555555555 = -1.590869007 Simplifying n + -0.5555555555 = -1.590869007 Reorder the terms: -0.5555555555 + n = -1.590869007 Solving -0.5555555555 + n = -1.590869007 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 = -1.590869007 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + n = -1.590869007 + 0.5555555555 n = -1.590869007 + 0.5555555555 Combine like terms: -1.590869007 + 0.5555555555 = -1.0353134515 n = -1.0353134515 Simplifying n = -1.0353134515Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.1464245625, -1.0353134515}
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