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Simplifying (2n + 1) * n = 1572 Reorder the terms: (1 + 2n) * n = 1572 Reorder the terms for easier multiplication: n(1 + 2n) = 1572 (1 * n + 2n * n) = 1572 (1n + 2n2) = 1572 Solving 1n + 2n2 = 1572 Solving for variable 'n'. Reorder the terms: -1572 + 1n + 2n2 = 1572 + -1572 Combine like terms: 1572 + -1572 = 0 -1572 + 1n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -786 + 0.5n + n2 = 0 Move the constant term to the right: Add '786' to each side of the equation. -786 + 0.5n + 786 + n2 = 0 + 786 Reorder the terms: -786 + 786 + 0.5n + n2 = 0 + 786 Combine like terms: -786 + 786 = 0 0 + 0.5n + n2 = 0 + 786 0.5n + n2 = 0 + 786 Combine like terms: 0 + 786 = 786 0.5n + n2 = 786 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 = 786 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 786 + 0.0625 Combine like terms: 786 + 0.0625 = 786.0625 0.0625 + 0.5n + n2 = 786.0625 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 786.0625 Calculate the square root of the right side: 28.036806166 Break this problem into two subproblems by setting (n + 0.25) equal to 28.036806166 and -28.036806166.Subproblem 1
n + 0.25 = 28.036806166 Simplifying n + 0.25 = 28.036806166 Reorder the terms: 0.25 + n = 28.036806166 Solving 0.25 + n = 28.036806166 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 = 28.036806166 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 28.036806166 + -0.25 n = 28.036806166 + -0.25 Combine like terms: 28.036806166 + -0.25 = 27.786806166 n = 27.786806166 Simplifying n = 27.786806166Subproblem 2
n + 0.25 = -28.036806166 Simplifying n + 0.25 = -28.036806166 Reorder the terms: 0.25 + n = -28.036806166 Solving 0.25 + n = -28.036806166 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 = -28.036806166 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -28.036806166 + -0.25 n = -28.036806166 + -0.25 Combine like terms: -28.036806166 + -0.25 = -28.286806166 n = -28.286806166 Simplifying n = -28.286806166Solution
The solution to the problem is based on the solutions from the subproblems. n = {27.786806166, -28.286806166}
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