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Simplifying (2n + 1) * n = 355 Reorder the terms: (1 + 2n) * n = 355 Reorder the terms for easier multiplication: n(1 + 2n) = 355 (1 * n + 2n * n) = 355 (1n + 2n2) = 355 Solving 1n + 2n2 = 355 Solving for variable 'n'. Reorder the terms: -355 + 1n + 2n2 = 355 + -355 Combine like terms: 355 + -355 = 0 -355 + 1n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -177.5 + 0.5n + n2 = 0 Move the constant term to the right: Add '177.5' to each side of the equation. -177.5 + 0.5n + 177.5 + n2 = 0 + 177.5 Reorder the terms: -177.5 + 177.5 + 0.5n + n2 = 0 + 177.5 Combine like terms: -177.5 + 177.5 = 0.0 0.0 + 0.5n + n2 = 0 + 177.5 0.5n + n2 = 0 + 177.5 Combine like terms: 0 + 177.5 = 177.5 0.5n + n2 = 177.5 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 177.5 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 177.5 + 0.0625 Combine like terms: 177.5 + 0.0625 = 177.5625 0.0625 + 0.5n + n2 = 177.5625 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 177.5625 Calculate the square root of the right side: 13.325257971 Break this problem into two subproblems by setting (n + 0.25) equal to 13.325257971 and -13.325257971.Subproblem 1
n + 0.25 = 13.325257971 Simplifying n + 0.25 = 13.325257971 Reorder the terms: 0.25 + n = 13.325257971 Solving 0.25 + n = 13.325257971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 13.325257971 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 13.325257971 + -0.25 n = 13.325257971 + -0.25 Combine like terms: 13.325257971 + -0.25 = 13.075257971 n = 13.075257971 Simplifying n = 13.075257971Subproblem 2
n + 0.25 = -13.325257971 Simplifying n + 0.25 = -13.325257971 Reorder the terms: 0.25 + n = -13.325257971 Solving 0.25 + n = -13.325257971 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -13.325257971 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -13.325257971 + -0.25 n = -13.325257971 + -0.25 Combine like terms: -13.325257971 + -0.25 = -13.575257971 n = -13.575257971 Simplifying n = -13.575257971Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.075257971, -13.575257971}
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