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Simplifying (8 * 2)(5 + -1n) = n * n + -2 Multiply 8 * 2 (16)(5 + -1n) = n * n + -2 (5 * 16 + -1n * 16) = n * n + -2 (80 + -16n) = n * n + -2 Multiply n * n 80 + -16n = n2 + -2 Reorder the terms: 80 + -16n = -2 + n2 Solving 80 + -16n = -2 + n2 Solving for variable 'n'. Reorder the terms: 80 + 2 + -16n + -1n2 = -2 + n2 + 2 + -1n2 Combine like terms: 80 + 2 = 82 82 + -16n + -1n2 = -2 + n2 + 2 + -1n2 Reorder the terms: 82 + -16n + -1n2 = -2 + 2 + n2 + -1n2 Combine like terms: -2 + 2 = 0 82 + -16n + -1n2 = 0 + n2 + -1n2 82 + -16n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 82 + -16n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -82 + 16n + n2 = 0 Move the constant term to the right: Add '82' to each side of the equation. -82 + 16n + 82 + n2 = 0 + 82 Reorder the terms: -82 + 82 + 16n + n2 = 0 + 82 Combine like terms: -82 + 82 = 0 0 + 16n + n2 = 0 + 82 16n + n2 = 0 + 82 Combine like terms: 0 + 82 = 82 16n + n2 = 82 The n term is 16n. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16n + 64 + n2 = 82 + 64 Reorder the terms: 64 + 16n + n2 = 82 + 64 Combine like terms: 82 + 64 = 146 64 + 16n + n2 = 146 Factor a perfect square on the left side: (n + 8)(n + 8) = 146 Calculate the square root of the right side: 12.083045974 Break this problem into two subproblems by setting (n + 8) equal to 12.083045974 and -12.083045974.Subproblem 1
n + 8 = 12.083045974 Simplifying n + 8 = 12.083045974 Reorder the terms: 8 + n = 12.083045974 Solving 8 + n = 12.083045974 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 12.083045974 + -8 Combine like terms: 8 + -8 = 0 0 + n = 12.083045974 + -8 n = 12.083045974 + -8 Combine like terms: 12.083045974 + -8 = 4.083045974 n = 4.083045974 Simplifying n = 4.083045974Subproblem 2
n + 8 = -12.083045974 Simplifying n + 8 = -12.083045974 Reorder the terms: 8 + n = -12.083045974 Solving 8 + n = -12.083045974 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -12.083045974 + -8 Combine like terms: 8 + -8 = 0 0 + n = -12.083045974 + -8 n = -12.083045974 + -8 Combine like terms: -12.083045974 + -8 = -20.083045974 n = -20.083045974 Simplifying n = -20.083045974Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.083045974, -20.083045974}
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