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Simplifying -0.16666666 = n2 + n Reorder the terms: -0.16666666 = n + n2 Solving -0.16666666 = n + n2 Solving for variable 'n'. Reorder the terms: -0.16666666 + -1n + -1n2 = n + -1n + n2 + -1n2 Combine like terms: n + -1n = 0 -0.16666666 + -1n + -1n2 = 0 + n2 + -1n2 -0.16666666 + -1n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 -0.16666666 + -1n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.16666666 + n + n2 = 0 Move the constant term to the right: Add '-0.16666666' to each side of the equation. 0.16666666 + n + -0.16666666 + n2 = 0 + -0.16666666 Reorder the terms: 0.16666666 + -0.16666666 + n + n2 = 0 + -0.16666666 Combine like terms: 0.16666666 + -0.16666666 = 0.00000000 0.00000000 + n + n2 = 0 + -0.16666666 n + n2 = 0 + -0.16666666 Combine like terms: 0 + -0.16666666 = -0.16666666 n + n2 = -0.16666666 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = -0.16666666 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = -0.16666666 + 0.25 Combine like terms: -0.16666666 + 0.25 = 0.08333334 1.25 + n2 = 0.08333334 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.08333334 Calculate the square root of the right side: 0.288675146 Break this problem into two subproblems by setting (n + 0.5) equal to 0.288675146 and -0.288675146.Subproblem 1
n + 0.5 = 0.288675146 Simplifying n + 0.5 = 0.288675146 Reorder the terms: 0.5 + n = 0.288675146 Solving 0.5 + n = 0.288675146 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.288675146 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.288675146 + -0.5 n = 0.288675146 + -0.5 Combine like terms: 0.288675146 + -0.5 = -0.211324854 n = -0.211324854 Simplifying n = -0.211324854Subproblem 2
n + 0.5 = -0.288675146 Simplifying n + 0.5 = -0.288675146 Reorder the terms: 0.5 + n = -0.288675146 Solving 0.5 + n = -0.288675146 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.288675146 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.288675146 + -0.5 n = -0.288675146 + -0.5 Combine like terms: -0.288675146 + -0.5 = -0.788675146 n = -0.788675146 Simplifying n = -0.788675146Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.211324854, -0.788675146}
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