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Simplifying n2 + 6n + -108 = 0 Reorder the terms: -108 + 6n + n2 = 0 Solving -108 + 6n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '108' to each side of the equation. -108 + 6n + 108 + n2 = 0 + 108 Reorder the terms: -108 + 108 + 6n + n2 = 0 + 108 Combine like terms: -108 + 108 = 0 0 + 6n + n2 = 0 + 108 6n + n2 = 0 + 108 Combine like terms: 0 + 108 = 108 6n + n2 = 108 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 = 108 + 9 Reorder the terms: 9 + 6n + n2 = 108 + 9 Combine like terms: 108 + 9 = 117 9 + 6n + n2 = 117 Factor a perfect square on the left side: (n + 3)(n + 3) = 117 Calculate the square root of the right side: 10.816653826 Break this problem into two subproblems by setting (n + 3) equal to 10.816653826 and -10.816653826.Subproblem 1
n + 3 = 10.816653826 Simplifying n + 3 = 10.816653826 Reorder the terms: 3 + n = 10.816653826 Solving 3 + n = 10.816653826 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 = 10.816653826 + -3 Combine like terms: 3 + -3 = 0 0 + n = 10.816653826 + -3 n = 10.816653826 + -3 Combine like terms: 10.816653826 + -3 = 7.816653826 n = 7.816653826 Simplifying n = 7.816653826Subproblem 2
n + 3 = -10.816653826 Simplifying n + 3 = -10.816653826 Reorder the terms: 3 + n = -10.816653826 Solving 3 + n = -10.816653826 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 = -10.816653826 + -3 Combine like terms: 3 + -3 = 0 0 + n = -10.816653826 + -3 n = -10.816653826 + -3 Combine like terms: -10.816653826 + -3 = -13.816653826 n = -13.816653826 Simplifying n = -13.816653826Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.816653826, -13.816653826}
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