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Simplifying 6n2 + -18n + -13 = 0 Reorder the terms: -13 + -18n + 6n2 = 0 Solving -13 + -18n + 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'. -2.166666667 + -3n + n2 = 0 Move the constant term to the right: Add '2.166666667' to each side of the equation. -2.166666667 + -3n + 2.166666667 + n2 = 0 + 2.166666667 Reorder the terms: -2.166666667 + 2.166666667 + -3n + n2 = 0 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + -3n + n2 = 0 + 2.166666667 -3n + n2 = 0 + 2.166666667 Combine like terms: 0 + 2.166666667 = 2.166666667 -3n + n2 = 2.166666667 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 2.166666667 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 2.166666667 + 2.25 Combine like terms: 2.166666667 + 2.25 = 4.416666667 2.25 + -3n + n2 = 4.416666667 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 4.416666667 Calculate the square root of the right side: 2.101586702 Break this problem into two subproblems by setting (n + -1.5) equal to 2.101586702 and -2.101586702.Subproblem 1
n + -1.5 = 2.101586702 Simplifying n + -1.5 = 2.101586702 Reorder the terms: -1.5 + n = 2.101586702 Solving -1.5 + n = 2.101586702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 2.101586702 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.101586702 + 1.5 n = 2.101586702 + 1.5 Combine like terms: 2.101586702 + 1.5 = 3.601586702 n = 3.601586702 Simplifying n = 3.601586702Subproblem 2
n + -1.5 = -2.101586702 Simplifying n + -1.5 = -2.101586702 Reorder the terms: -1.5 + n = -2.101586702 Solving -1.5 + n = -2.101586702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -2.101586702 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.101586702 + 1.5 n = -2.101586702 + 1.5 Combine like terms: -2.101586702 + 1.5 = -0.601586702 n = -0.601586702 Simplifying n = -0.601586702Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.601586702, -0.601586702}
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