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Simplifying 2n2 + 6n + 25 = 5184 Reorder the terms: 25 + 6n + 2n2 = 5184 Solving 25 + 6n + 2n2 = 5184 Solving for variable 'n'. Reorder the terms: 25 + -5184 + 6n + 2n2 = 5184 + -5184 Combine like terms: 25 + -5184 = -5159 -5159 + 6n + 2n2 = 5184 + -5184 Combine like terms: 5184 + -5184 = 0 -5159 + 6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2579.5 + 3n + n2 = 0 Move the constant term to the right: Add '2579.5' to each side of the equation. -2579.5 + 3n + 2579.5 + n2 = 0 + 2579.5 Reorder the terms: -2579.5 + 2579.5 + 3n + n2 = 0 + 2579.5 Combine like terms: -2579.5 + 2579.5 = 0.0 0.0 + 3n + n2 = 0 + 2579.5 3n + n2 = 0 + 2579.5 Combine like terms: 0 + 2579.5 = 2579.5 3n + n2 = 2579.5 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 2579.5 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 2579.5 + 2.25 Combine like terms: 2579.5 + 2.25 = 2581.75 2.25 + 3n + n2 = 2581.75 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 2581.75 Calculate the square root of the right side: 50.810924022 Break this problem into two subproblems by setting (n + 1.5) equal to 50.810924022 and -50.810924022.Subproblem 1
n + 1.5 = 50.810924022 Simplifying n + 1.5 = 50.810924022 Reorder the terms: 1.5 + n = 50.810924022 Solving 1.5 + n = 50.810924022 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 50.810924022 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 50.810924022 + -1.5 n = 50.810924022 + -1.5 Combine like terms: 50.810924022 + -1.5 = 49.310924022 n = 49.310924022 Simplifying n = 49.310924022Subproblem 2
n + 1.5 = -50.810924022 Simplifying n + 1.5 = -50.810924022 Reorder the terms: 1.5 + n = -50.810924022 Solving 1.5 + n = -50.810924022 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -50.810924022 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -50.810924022 + -1.5 n = -50.810924022 + -1.5 Combine like terms: -50.810924022 + -1.5 = -52.310924022 n = -52.310924022 Simplifying n = -52.310924022Solution
The solution to the problem is based on the solutions from the subproblems. n = {49.310924022, -52.310924022}
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