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Simplifying n2 + n = 198 Reorder the terms: n + n2 = 198 Solving n + n2 = 198 Solving for variable 'n'. Reorder the terms: -198 + n + n2 = 198 + -198 Combine like terms: 198 + -198 = 0 -198 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '198' to each side of the equation. -198 + n + 198 + n2 = 0 + 198 Reorder the terms: -198 + 198 + n + n2 = 0 + 198 Combine like terms: -198 + 198 = 0 0 + n + n2 = 0 + 198 n + n2 = 0 + 198 Combine like terms: 0 + 198 = 198 n + n2 = 198 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 = 198 + 0.25 Reorder the terms: 0.25 + n + n2 = 198 + 0.25 Combine like terms: 198 + 0.25 = 198.25 0.25 + n + n2 = 198.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 198.25 Calculate the square root of the right side: 14.08012784 Break this problem into two subproblems by setting (n + 0.5) equal to 14.08012784 and -14.08012784.Subproblem 1
n + 0.5 = 14.08012784 Simplifying n + 0.5 = 14.08012784 Reorder the terms: 0.5 + n = 14.08012784 Solving 0.5 + n = 14.08012784 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 = 14.08012784 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 14.08012784 + -0.5 n = 14.08012784 + -0.5 Combine like terms: 14.08012784 + -0.5 = 13.58012784 n = 13.58012784 Simplifying n = 13.58012784Subproblem 2
n + 0.5 = -14.08012784 Simplifying n + 0.5 = -14.08012784 Reorder the terms: 0.5 + n = -14.08012784 Solving 0.5 + n = -14.08012784 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 = -14.08012784 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -14.08012784 + -0.5 n = -14.08012784 + -0.5 Combine like terms: -14.08012784 + -0.5 = -14.58012784 n = -14.58012784 Simplifying n = -14.58012784Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.58012784, -14.58012784}
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