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Simplifying n2 + 6n + -1000 = 0 Reorder the terms: -1000 + 6n + n2 = 0 Solving -1000 + 6n + 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 + 6n + 1000 + n2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 6n + n2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 6n + n2 = 0 + 1000 6n + n2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 6n + n2 = 1000 The n term is 6n. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6n + 9 + n2 = 1000 + 9 Reorder the terms: 9 + 6n + n2 = 1000 + 9 Combine like terms: 1000 + 9 = 1009 9 + 6n + n2 = 1009 Factor a perfect square on the left side: (n + 3)(n + 3) = 1009 Calculate the square root of the right side: 31.764760349 Break this problem into two subproblems by setting (n + 3) equal to 31.764760349 and -31.764760349.Subproblem 1
n + 3 = 31.764760349 Simplifying n + 3 = 31.764760349 Reorder the terms: 3 + n = 31.764760349 Solving 3 + n = 31.764760349 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 31.764760349 + -3 Combine like terms: 3 + -3 = 0 0 + n = 31.764760349 + -3 n = 31.764760349 + -3 Combine like terms: 31.764760349 + -3 = 28.764760349 n = 28.764760349 Simplifying n = 28.764760349Subproblem 2
n + 3 = -31.764760349 Simplifying n + 3 = -31.764760349 Reorder the terms: 3 + n = -31.764760349 Solving 3 + n = -31.764760349 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -31.764760349 + -3 Combine like terms: 3 + -3 = 0 0 + n = -31.764760349 + -3 n = -31.764760349 + -3 Combine like terms: -31.764760349 + -3 = -34.764760349 n = -34.764760349 Simplifying n = -34.764760349Solution
The solution to the problem is based on the solutions from the subproblems. n = {28.764760349, -34.764760349}
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