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Simplifying n(n + 6) = 176 Reorder the terms: n(6 + n) = 176 (6 * n + n * n) = 176 (6n + n2) = 176 Solving 6n + n2 = 176 Solving for variable 'n'. Reorder the terms: -176 + 6n + n2 = 176 + -176 Combine like terms: 176 + -176 = 0 -176 + 6n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '176' to each side of the equation. -176 + 6n + 176 + n2 = 0 + 176 Reorder the terms: -176 + 176 + 6n + n2 = 0 + 176 Combine like terms: -176 + 176 = 0 0 + 6n + n2 = 0 + 176 6n + n2 = 0 + 176 Combine like terms: 0 + 176 = 176 6n + n2 = 176 The n term is 6n. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6n + 9 + n2 = 176 + 9 Reorder the terms: 9 + 6n + n2 = 176 + 9 Combine like terms: 176 + 9 = 185 9 + 6n + n2 = 185 Factor a perfect square on the left side: (n + 3)(n + 3) = 185 Calculate the square root of the right side: 13.601470509 Break this problem into two subproblems by setting (n + 3) equal to 13.601470509 and -13.601470509.Subproblem 1
n + 3 = 13.601470509 Simplifying n + 3 = 13.601470509 Reorder the terms: 3 + n = 13.601470509 Solving 3 + n = 13.601470509 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 13.601470509 + -3 Combine like terms: 3 + -3 = 0 0 + n = 13.601470509 + -3 n = 13.601470509 + -3 Combine like terms: 13.601470509 + -3 = 10.601470509 n = 10.601470509 Simplifying n = 10.601470509Subproblem 2
n + 3 = -13.601470509 Simplifying n + 3 = -13.601470509 Reorder the terms: 3 + n = -13.601470509 Solving 3 + n = -13.601470509 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -13.601470509 + -3 Combine like terms: 3 + -3 = 0 0 + n = -13.601470509 + -3 n = -13.601470509 + -3 Combine like terms: -13.601470509 + -3 = -16.601470509 n = -16.601470509 Simplifying n = -16.601470509Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.601470509, -16.601470509}
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