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Simplifying n2 + 2n + -68 = 7 Reorder the terms: -68 + 2n + n2 = 7 Solving -68 + 2n + n2 = 7 Solving for variable 'n'. Reorder the terms: -68 + -7 + 2n + n2 = 7 + -7 Combine like terms: -68 + -7 = -75 -75 + 2n + n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -75 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '75' to each side of the equation. -75 + 2n + 75 + n2 = 0 + 75 Reorder the terms: -75 + 75 + 2n + n2 = 0 + 75 Combine like terms: -75 + 75 = 0 0 + 2n + n2 = 0 + 75 2n + n2 = 0 + 75 Combine like terms: 0 + 75 = 75 2n + n2 = 75 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 = 75 + 1 Reorder the terms: 1 + 2n + n2 = 75 + 1 Combine like terms: 75 + 1 = 76 1 + 2n + n2 = 76 Factor a perfect square on the left side: (n + 1)(n + 1) = 76 Calculate the square root of the right side: 8.717797887 Break this problem into two subproblems by setting (n + 1) equal to 8.717797887 and -8.717797887.Subproblem 1
n + 1 = 8.717797887 Simplifying n + 1 = 8.717797887 Reorder the terms: 1 + n = 8.717797887 Solving 1 + n = 8.717797887 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 = 8.717797887 + -1 Combine like terms: 1 + -1 = 0 0 + n = 8.717797887 + -1 n = 8.717797887 + -1 Combine like terms: 8.717797887 + -1 = 7.717797887 n = 7.717797887 Simplifying n = 7.717797887Subproblem 2
n + 1 = -8.717797887 Simplifying n + 1 = -8.717797887 Reorder the terms: 1 + n = -8.717797887 Solving 1 + n = -8.717797887 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 = -8.717797887 + -1 Combine like terms: 1 + -1 = 0 0 + n = -8.717797887 + -1 n = -8.717797887 + -1 Combine like terms: -8.717797887 + -1 = -9.717797887 n = -9.717797887 Simplifying n = -9.717797887Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.717797887, -9.717797887}
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