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Simplifying n2 + 9n + -1000 = 0 Reorder the terms: -1000 + 9n + n2 = 0 Solving -1000 + 9n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 9n + 1000 + n2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 9n + n2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 9n + n2 = 0 + 1000 9n + n2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 9n + n2 = 1000 The n term is 9n. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9n + 20.25 + n2 = 1000 + 20.25 Reorder the terms: 20.25 + 9n + n2 = 1000 + 20.25 Combine like terms: 1000 + 20.25 = 1020.25 20.25 + 9n + n2 = 1020.25 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 1020.25 Calculate the square root of the right side: 31.941352507 Break this problem into two subproblems by setting (n + 4.5) equal to 31.941352507 and -31.941352507.Subproblem 1
n + 4.5 = 31.941352507 Simplifying n + 4.5 = 31.941352507 Reorder the terms: 4.5 + n = 31.941352507 Solving 4.5 + n = 31.941352507 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 31.941352507 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 31.941352507 + -4.5 n = 31.941352507 + -4.5 Combine like terms: 31.941352507 + -4.5 = 27.441352507 n = 27.441352507 Simplifying n = 27.441352507Subproblem 2
n + 4.5 = -31.941352507 Simplifying n + 4.5 = -31.941352507 Reorder the terms: 4.5 + n = -31.941352507 Solving 4.5 + n = -31.941352507 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -31.941352507 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -31.941352507 + -4.5 n = -31.941352507 + -4.5 Combine like terms: -31.941352507 + -4.5 = -36.441352507 n = -36.441352507 Simplifying n = -36.441352507Solution
The solution to the problem is based on the solutions from the subproblems. n = {27.441352507, -36.441352507}
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