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Simplifying 16n2 + -48n = -29 Reorder the terms: -48n + 16n2 = -29 Solving -48n + 16n2 = -29 Solving for variable 'n'. Reorder the terms: 29 + -48n + 16n2 = -29 + 29 Combine like terms: -29 + 29 = 0 29 + -48n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.8125 + -3n + n2 = 0 Move the constant term to the right: Add '-1.8125' to each side of the equation. 1.8125 + -3n + -1.8125 + n2 = 0 + -1.8125 Reorder the terms: 1.8125 + -1.8125 + -3n + n2 = 0 + -1.8125 Combine like terms: 1.8125 + -1.8125 = 0.0000 0.0000 + -3n + n2 = 0 + -1.8125 -3n + n2 = 0 + -1.8125 Combine like terms: 0 + -1.8125 = -1.8125 -3n + n2 = -1.8125 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 = -1.8125 + 2.25 Reorder the terms: 2.25 + -3n + n2 = -1.8125 + 2.25 Combine like terms: -1.8125 + 2.25 = 0.4375 2.25 + -3n + n2 = 0.4375 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (n + -1.5) equal to 0.661437828 and -0.661437828.Subproblem 1
n + -1.5 = 0.661437828 Simplifying n + -1.5 = 0.661437828 Reorder the terms: -1.5 + n = 0.661437828 Solving -1.5 + n = 0.661437828 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 = 0.661437828 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 0.661437828 + 1.5 n = 0.661437828 + 1.5 Combine like terms: 0.661437828 + 1.5 = 2.161437828 n = 2.161437828 Simplifying n = 2.161437828Subproblem 2
n + -1.5 = -0.661437828 Simplifying n + -1.5 = -0.661437828 Reorder the terms: -1.5 + n = -0.661437828 Solving -1.5 + n = -0.661437828 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 = -0.661437828 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -0.661437828 + 1.5 n = -0.661437828 + 1.5 Combine like terms: -0.661437828 + 1.5 = 0.838562172 n = 0.838562172 Simplifying n = 0.838562172Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.161437828, 0.838562172}
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