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Simplifying n2 + -2n + -15 = 102 Reorder the terms: -15 + -2n + n2 = 102 Solving -15 + -2n + n2 = 102 Solving for variable 'n'. Reorder the terms: -15 + -102 + -2n + n2 = 102 + -102 Combine like terms: -15 + -102 = -117 -117 + -2n + n2 = 102 + -102 Combine like terms: 102 + -102 = 0 -117 + -2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '117' to each side of the equation. -117 + -2n + 117 + n2 = 0 + 117 Reorder the terms: -117 + 117 + -2n + n2 = 0 + 117 Combine like terms: -117 + 117 = 0 0 + -2n + n2 = 0 + 117 -2n + n2 = 0 + 117 Combine like terms: 0 + 117 = 117 -2n + n2 = 117 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 117 + 1 Reorder the terms: 1 + -2n + n2 = 117 + 1 Combine like terms: 117 + 1 = 118 1 + -2n + n2 = 118 Factor a perfect square on the left side: (n + -1)(n + -1) = 118 Calculate the square root of the right side: 10.862780491 Break this problem into two subproblems by setting (n + -1) equal to 10.862780491 and -10.862780491.Subproblem 1
n + -1 = 10.862780491 Simplifying n + -1 = 10.862780491 Reorder the terms: -1 + n = 10.862780491 Solving -1 + n = 10.862780491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 10.862780491 + 1 Combine like terms: -1 + 1 = 0 0 + n = 10.862780491 + 1 n = 10.862780491 + 1 Combine like terms: 10.862780491 + 1 = 11.862780491 n = 11.862780491 Simplifying n = 11.862780491Subproblem 2
n + -1 = -10.862780491 Simplifying n + -1 = -10.862780491 Reorder the terms: -1 + n = -10.862780491 Solving -1 + n = -10.862780491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -10.862780491 + 1 Combine like terms: -1 + 1 = 0 0 + n = -10.862780491 + 1 n = -10.862780491 + 1 Combine like terms: -10.862780491 + 1 = -9.862780491 n = -9.862780491 Simplifying n = -9.862780491Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.862780491, -9.862780491}
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