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Simplifying -6n2 + 9n + 8 = 0 Reorder the terms: 8 + 9n + -6n2 = 0 Solving 8 + 9n + -6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.333333333 + -1.5n + n2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -1.5n + 1.333333333 + n2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -1.5n + n2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -1.5n + n2 = 0 + 1.333333333 -1.5n + n2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -1.5n + n2 = 1.333333333 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 = 1.333333333 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 1.333333333 + 0.5625 Combine like terms: 1.333333333 + 0.5625 = 1.895833333 0.5625 + -1.5n + n2 = 1.895833333 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 1.895833333 Calculate the square root of the right side: 1.376892637 Break this problem into two subproblems by setting (n + -0.75) equal to 1.376892637 and -1.376892637.Subproblem 1
n + -0.75 = 1.376892637 Simplifying n + -0.75 = 1.376892637 Reorder the terms: -0.75 + n = 1.376892637 Solving -0.75 + n = 1.376892637 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 = 1.376892637 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.376892637 + 0.75 n = 1.376892637 + 0.75 Combine like terms: 1.376892637 + 0.75 = 2.126892637 n = 2.126892637 Simplifying n = 2.126892637Subproblem 2
n + -0.75 = -1.376892637 Simplifying n + -0.75 = -1.376892637 Reorder the terms: -0.75 + n = -1.376892637 Solving -0.75 + n = -1.376892637 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 = -1.376892637 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.376892637 + 0.75 n = -1.376892637 + 0.75 Combine like terms: -1.376892637 + 0.75 = -0.626892637 n = -0.626892637 Simplifying n = -0.626892637Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.126892637, -0.626892637}
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