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Simplifying 8n2 + -12n = -1 Reorder the terms: -12n + 8n2 = -1 Solving -12n + 8n2 = -1 Solving for variable 'n'. Reorder the terms: 1 + -12n + 8n2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + -12n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + -1.5n + n2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.5n + -0.125 + n2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.5n + n2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.5n + n2 = 0 + -0.125 -1.5n + n2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.5n + n2 = -0.125 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = -0.125 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = -0.125 + 0.5625 Combine like terms: -0.125 + 0.5625 = 0.4375 0.5625 + -1.5n + n2 = 0.4375 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (n + -0.75) equal to 0.661437828 and -0.661437828.Subproblem 1
n + -0.75 = 0.661437828 Simplifying n + -0.75 = 0.661437828 Reorder the terms: -0.75 + n = 0.661437828 Solving -0.75 + n = 0.661437828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 0.661437828 + 0.75 n = 0.661437828 + 0.75 Combine like terms: 0.661437828 + 0.75 = 1.411437828 n = 1.411437828 Simplifying n = 1.411437828Subproblem 2
n + -0.75 = -0.661437828 Simplifying n + -0.75 = -0.661437828 Reorder the terms: -0.75 + n = -0.661437828 Solving -0.75 + n = -0.661437828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -0.661437828 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -0.661437828 + 0.75 n = -0.661437828 + 0.75 Combine like terms: -0.661437828 + 0.75 = 0.088562172 n = 0.088562172 Simplifying n = 0.088562172Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.411437828, 0.088562172}
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