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Simplifying -1.25n2 + 41.25n = 162 Reorder the terms: 41.25n + -1.25n2 = 162 Solving 41.25n + -1.25n2 = 162 Solving for variable 'n'. Reorder the terms: -162 + 41.25n + -1.25n2 = 162 + -162 Combine like terms: 162 + -162 = 0 -162 + 41.25n + -1.25n2 = 0 Begin completing the square. Divide all terms by -1.25 the coefficient of the squared term: Divide each side by '-1.25'. 129.6 + -33n + n2 = 0 Move the constant term to the right: Add '-129.6' to each side of the equation. 129.6 + -33n + -129.6 + n2 = 0 + -129.6 Reorder the terms: 129.6 + -129.6 + -33n + n2 = 0 + -129.6 Combine like terms: 129.6 + -129.6 = 0.0 0.0 + -33n + n2 = 0 + -129.6 -33n + n2 = 0 + -129.6 Combine like terms: 0 + -129.6 = -129.6 -33n + n2 = -129.6 The n term is -33n. Take half its coefficient (-16.5). Square it (272.25) and add it to both sides. Add '272.25' to each side of the equation. -33n + 272.25 + n2 = -129.6 + 272.25 Reorder the terms: 272.25 + -33n + n2 = -129.6 + 272.25 Combine like terms: -129.6 + 272.25 = 142.65 272.25 + -33n + n2 = 142.65 Factor a perfect square on the left side: (n + -16.5)(n + -16.5) = 142.65 Calculate the square root of the right side: 11.943617542 Break this problem into two subproblems by setting (n + -16.5) equal to 11.943617542 and -11.943617542.Subproblem 1
n + -16.5 = 11.943617542 Simplifying n + -16.5 = 11.943617542 Reorder the terms: -16.5 + n = 11.943617542 Solving -16.5 + n = 11.943617542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '16.5' to each side of the equation. -16.5 + 16.5 + n = 11.943617542 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + n = 11.943617542 + 16.5 n = 11.943617542 + 16.5 Combine like terms: 11.943617542 + 16.5 = 28.443617542 n = 28.443617542 Simplifying n = 28.443617542Subproblem 2
n + -16.5 = -11.943617542 Simplifying n + -16.5 = -11.943617542 Reorder the terms: -16.5 + n = -11.943617542 Solving -16.5 + n = -11.943617542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '16.5' to each side of the equation. -16.5 + 16.5 + n = -11.943617542 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + n = -11.943617542 + 16.5 n = -11.943617542 + 16.5 Combine like terms: -11.943617542 + 16.5 = 4.556382458 n = 4.556382458 Simplifying n = 4.556382458Solution
The solution to the problem is based on the solutions from the subproblems. n = {28.443617542, 4.556382458}
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