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Simplifying (2n + 1) * n = 1255 Reorder the terms: (1 + 2n) * n = 1255 Reorder the terms for easier multiplication: n(1 + 2n) = 1255 (1 * n + 2n * n) = 1255 (1n + 2n2) = 1255 Solving 1n + 2n2 = 1255 Solving for variable 'n'. Reorder the terms: -1255 + 1n + 2n2 = 1255 + -1255 Combine like terms: 1255 + -1255 = 0 -1255 + 1n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -627.5 + 0.5n + n2 = 0 Move the constant term to the right: Add '627.5' to each side of the equation. -627.5 + 0.5n + 627.5 + n2 = 0 + 627.5 Reorder the terms: -627.5 + 627.5 + 0.5n + n2 = 0 + 627.5 Combine like terms: -627.5 + 627.5 = 0.0 0.0 + 0.5n + n2 = 0 + 627.5 0.5n + n2 = 0 + 627.5 Combine like terms: 0 + 627.5 = 627.5 0.5n + n2 = 627.5 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 = 627.5 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 627.5 + 0.0625 Combine like terms: 627.5 + 0.0625 = 627.5625 0.0625 + 0.5n + n2 = 627.5625 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 627.5625 Calculate the square root of the right side: 25.051197576 Break this problem into two subproblems by setting (n + 0.25) equal to 25.051197576 and -25.051197576.Subproblem 1
n + 0.25 = 25.051197576 Simplifying n + 0.25 = 25.051197576 Reorder the terms: 0.25 + n = 25.051197576 Solving 0.25 + n = 25.051197576 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 = 25.051197576 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 25.051197576 + -0.25 n = 25.051197576 + -0.25 Combine like terms: 25.051197576 + -0.25 = 24.801197576 n = 24.801197576 Simplifying n = 24.801197576Subproblem 2
n + 0.25 = -25.051197576 Simplifying n + 0.25 = -25.051197576 Reorder the terms: 0.25 + n = -25.051197576 Solving 0.25 + n = -25.051197576 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 = -25.051197576 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -25.051197576 + -0.25 n = -25.051197576 + -0.25 Combine like terms: -25.051197576 + -0.25 = -25.301197576 n = -25.301197576 Simplifying n = -25.301197576Solution
The solution to the problem is based on the solutions from the subproblems. n = {24.801197576, -25.301197576}
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