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Simplifying 3n2 + n + -1000 = 0 Reorder the terms: -1000 + n + 3n2 = 0 Solving -1000 + n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -333.3333333 + 0.3333333333n + n2 = 0 Move the constant term to the right: Add '333.3333333' to each side of the equation. -333.3333333 + 0.3333333333n + 333.3333333 + n2 = 0 + 333.3333333 Reorder the terms: -333.3333333 + 333.3333333 + 0.3333333333n + n2 = 0 + 333.3333333 Combine like terms: -333.3333333 + 333.3333333 = 0.0000000 0.0000000 + 0.3333333333n + n2 = 0 + 333.3333333 0.3333333333n + n2 = 0 + 333.3333333 Combine like terms: 0 + 333.3333333 = 333.3333333 0.3333333333n + n2 = 333.3333333 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333n + 0.25 + n2 = 333.3333333 + 0.25 Reorder the terms: 0.25 + 0.3333333333n + n2 = 333.3333333 + 0.25 Combine like terms: 333.3333333 + 0.25 = 333.5833333 0.25 + 0.3333333333n + n2 = 333.5833333 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 333.5833333 Calculate the square root of the right side: 18.264263831 Break this problem into two subproblems by setting (n + 0.5) equal to 18.264263831 and -18.264263831.Subproblem 1
n + 0.5 = 18.264263831 Simplifying n + 0.5 = 18.264263831 Reorder the terms: 0.5 + n = 18.264263831 Solving 0.5 + n = 18.264263831 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 18.264263831 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 18.264263831 + -0.5 n = 18.264263831 + -0.5 Combine like terms: 18.264263831 + -0.5 = 17.764263831 n = 17.764263831 Simplifying n = 17.764263831Subproblem 2
n + 0.5 = -18.264263831 Simplifying n + 0.5 = -18.264263831 Reorder the terms: 0.5 + n = -18.264263831 Solving 0.5 + n = -18.264263831 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -18.264263831 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -18.264263831 + -0.5 n = -18.264263831 + -0.5 Combine like terms: -18.264263831 + -0.5 = -18.764263831 n = -18.764263831 Simplifying n = -18.764263831Solution
The solution to the problem is based on the solutions from the subproblems. n = {17.764263831, -18.764263831}
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