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Simplifying (12 + n)(13 + n) = 200 Multiply (12 + n) * (13 + n) (12(13 + n) + n(13 + n)) = 200 ((13 * 12 + n * 12) + n(13 + n)) = 200 ((156 + 12n) + n(13 + n)) = 200 (156 + 12n + (13 * n + n * n)) = 200 (156 + 12n + (13n + n2)) = 200 Combine like terms: 12n + 13n = 25n (156 + 25n + n2) = 200 Solving 156 + 25n + n2 = 200 Solving for variable 'n'. Reorder the terms: 156 + -200 + 25n + n2 = 200 + -200 Combine like terms: 156 + -200 = -44 -44 + 25n + n2 = 200 + -200 Combine like terms: 200 + -200 = 0 -44 + 25n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '44' to each side of the equation. -44 + 25n + 44 + n2 = 0 + 44 Reorder the terms: -44 + 44 + 25n + n2 = 0 + 44 Combine like terms: -44 + 44 = 0 0 + 25n + n2 = 0 + 44 25n + n2 = 0 + 44 Combine like terms: 0 + 44 = 44 25n + n2 = 44 The n term is 25n. Take half its coefficient (12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. 25n + 156.25 + n2 = 44 + 156.25 Reorder the terms: 156.25 + 25n + n2 = 44 + 156.25 Combine like terms: 44 + 156.25 = 200.25 156.25 + 25n + n2 = 200.25 Factor a perfect square on the left side: (n + 12.5)(n + 12.5) = 200.25 Calculate the square root of the right side: 14.150971698 Break this problem into two subproblems by setting (n + 12.5) equal to 14.150971698 and -14.150971698.Subproblem 1
n + 12.5 = 14.150971698 Simplifying n + 12.5 = 14.150971698 Reorder the terms: 12.5 + n = 14.150971698 Solving 12.5 + n = 14.150971698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + n = 14.150971698 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + n = 14.150971698 + -12.5 n = 14.150971698 + -12.5 Combine like terms: 14.150971698 + -12.5 = 1.650971698 n = 1.650971698 Simplifying n = 1.650971698Subproblem 2
n + 12.5 = -14.150971698 Simplifying n + 12.5 = -14.150971698 Reorder the terms: 12.5 + n = -14.150971698 Solving 12.5 + n = -14.150971698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + n = -14.150971698 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + n = -14.150971698 + -12.5 n = -14.150971698 + -12.5 Combine like terms: -14.150971698 + -12.5 = -26.650971698 n = -26.650971698 Simplifying n = -26.650971698Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.650971698, -26.650971698}
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