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Simplifying n2 + n = 1000 Reorder the terms: n + n2 = 1000 Solving n + n2 = 1000 Solving for variable 'n'. Reorder the terms: -1000 + n + n2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + n + 1000 + n2 = 0 + 1000 Reorder the terms: -1000 + 1000 + n + n2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + n + n2 = 0 + 1000 n + n2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 n + n2 = 1000 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. n + 0.25 + n2 = 1000 + 0.25 Reorder the terms: 0.25 + n + n2 = 1000 + 0.25 Combine like terms: 1000 + 0.25 = 1000.25 0.25 + n + n2 = 1000.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 1000.25 Calculate the square root of the right side: 31.626729202 Break this problem into two subproblems by setting (n + 0.5) equal to 31.626729202 and -31.626729202.Subproblem 1
n + 0.5 = 31.626729202 Simplifying n + 0.5 = 31.626729202 Reorder the terms: 0.5 + n = 31.626729202 Solving 0.5 + n = 31.626729202 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 = 31.626729202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 31.626729202 + -0.5 n = 31.626729202 + -0.5 Combine like terms: 31.626729202 + -0.5 = 31.126729202 n = 31.126729202 Simplifying n = 31.126729202Subproblem 2
n + 0.5 = -31.626729202 Simplifying n + 0.5 = -31.626729202 Reorder the terms: 0.5 + n = -31.626729202 Solving 0.5 + n = -31.626729202 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 = -31.626729202 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -31.626729202 + -0.5 n = -31.626729202 + -0.5 Combine like terms: -31.626729202 + -0.5 = -32.126729202 n = -32.126729202 Simplifying n = -32.126729202Solution
The solution to the problem is based on the solutions from the subproblems. n = {31.126729202, -32.126729202}
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