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Simplifying 2n(n + 3) = 23 Reorder the terms: 2n(3 + n) = 23 (3 * 2n + n * 2n) = 23 (6n + 2n2) = 23 Solving 6n + 2n2 = 23 Solving for variable 'n'. Reorder the terms: -23 + 6n + 2n2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11.5 + 3n + n2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 3n + 11.5 + n2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 3n + n2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 3n + n2 = 0 + 11.5 3n + n2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 3n + n2 = 11.5 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 = 11.5 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 11.5 + 2.25 Combine like terms: 11.5 + 2.25 = 13.75 2.25 + 3n + n2 = 13.75 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 13.75 Calculate the square root of the right side: 3.708099244 Break this problem into two subproblems by setting (n + 1.5) equal to 3.708099244 and -3.708099244.Subproblem 1
n + 1.5 = 3.708099244 Simplifying n + 1.5 = 3.708099244 Reorder the terms: 1.5 + n = 3.708099244 Solving 1.5 + n = 3.708099244 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 = 3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 3.708099244 + -1.5 n = 3.708099244 + -1.5 Combine like terms: 3.708099244 + -1.5 = 2.208099244 n = 2.208099244 Simplifying n = 2.208099244Subproblem 2
n + 1.5 = -3.708099244 Simplifying n + 1.5 = -3.708099244 Reorder the terms: 1.5 + n = -3.708099244 Solving 1.5 + n = -3.708099244 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 = -3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -3.708099244 + -1.5 n = -3.708099244 + -1.5 Combine like terms: -3.708099244 + -1.5 = -5.208099244 n = -5.208099244 Simplifying n = -5.208099244Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.208099244, -5.208099244}
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