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Simplifying 2n(n + -3) = 4n + 1 Reorder the terms: 2n(-3 + n) = 4n + 1 (-3 * 2n + n * 2n) = 4n + 1 (-6n + 2n2) = 4n + 1 Reorder the terms: -6n + 2n2 = 1 + 4n Solving -6n + 2n2 = 1 + 4n Solving for variable 'n'. Reorder the terms: -1 + -6n + -4n + 2n2 = 1 + 4n + -1 + -4n Combine like terms: -6n + -4n = -10n -1 + -10n + 2n2 = 1 + 4n + -1 + -4n Reorder the terms: -1 + -10n + 2n2 = 1 + -1 + 4n + -4n Combine like terms: 1 + -1 = 0 -1 + -10n + 2n2 = 0 + 4n + -4n -1 + -10n + 2n2 = 4n + -4n Combine like terms: 4n + -4n = 0 -1 + -10n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -5n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -5n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -5n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -5n + n2 = 0 + 0.5 -5n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -5n + n2 = 0.5 The n term is -5n. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5n + 6.25 + n2 = 0.5 + 6.25 Reorder the terms: 6.25 + -5n + n2 = 0.5 + 6.25 Combine like terms: 0.5 + 6.25 = 6.75 6.25 + -5n + n2 = 6.75 Factor a perfect square on the left side: (n + -2.5)(n + -2.5) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (n + -2.5) equal to 2.598076211 and -2.598076211.Subproblem 1
n + -2.5 = 2.598076211 Simplifying n + -2.5 = 2.598076211 Reorder the terms: -2.5 + n = 2.598076211 Solving -2.5 + n = 2.598076211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 2.598076211 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 2.598076211 + 2.5 n = 2.598076211 + 2.5 Combine like terms: 2.598076211 + 2.5 = 5.098076211 n = 5.098076211 Simplifying n = 5.098076211Subproblem 2
n + -2.5 = -2.598076211 Simplifying n + -2.5 = -2.598076211 Reorder the terms: -2.5 + n = -2.598076211 Solving -2.5 + n = -2.598076211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -2.598076211 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -2.598076211 + 2.5 n = -2.598076211 + 2.5 Combine like terms: -2.598076211 + 2.5 = -0.098076211 n = -0.098076211 Simplifying n = -0.098076211Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.098076211, -0.098076211}
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