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Simplifying n2 + -18n + -105 = 0 Reorder the terms: -105 + -18n + n2 = 0 Solving -105 + -18n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '105' to each side of the equation. -105 + -18n + 105 + n2 = 0 + 105 Reorder the terms: -105 + 105 + -18n + n2 = 0 + 105 Combine like terms: -105 + 105 = 0 0 + -18n + n2 = 0 + 105 -18n + n2 = 0 + 105 Combine like terms: 0 + 105 = 105 -18n + n2 = 105 The n term is -18n. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18n + 81 + n2 = 105 + 81 Reorder the terms: 81 + -18n + n2 = 105 + 81 Combine like terms: 105 + 81 = 186 81 + -18n + n2 = 186 Factor a perfect square on the left side: (n + -9)(n + -9) = 186 Calculate the square root of the right side: 13.638181697 Break this problem into two subproblems by setting (n + -9) equal to 13.638181697 and -13.638181697.Subproblem 1
n + -9 = 13.638181697 Simplifying n + -9 = 13.638181697 Reorder the terms: -9 + n = 13.638181697 Solving -9 + n = 13.638181697 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 13.638181697 + 9 Combine like terms: -9 + 9 = 0 0 + n = 13.638181697 + 9 n = 13.638181697 + 9 Combine like terms: 13.638181697 + 9 = 22.638181697 n = 22.638181697 Simplifying n = 22.638181697Subproblem 2
n + -9 = -13.638181697 Simplifying n + -9 = -13.638181697 Reorder the terms: -9 + n = -13.638181697 Solving -9 + n = -13.638181697 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -13.638181697 + 9 Combine like terms: -9 + 9 = 0 0 + n = -13.638181697 + 9 n = -13.638181697 + 9 Combine like terms: -13.638181697 + 9 = -4.638181697 n = -4.638181697 Simplifying n = -4.638181697Solution
The solution to the problem is based on the solutions from the subproblems. n = {22.638181697, -4.638181697}
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