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Simplifying n2 + 3n + -12 + 6 = 0 Reorder the terms: -12 + 6 + 3n + n2 = 0 Combine like terms: -12 + 6 = -6 -6 + 3n + n2 = 0 Solving -6 + 3n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + 3n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + 3n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 3n + n2 = 0 + 6 3n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 3n + n2 = 6 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 = 6 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 6 + 2.25 Combine like terms: 6 + 2.25 = 8.25 2.25 + 3n + n2 = 8.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (n + 1.5) equal to 2.872281323 and -2.872281323.Subproblem 1
n + 1.5 = 2.872281323 Simplifying n + 1.5 = 2.872281323 Reorder the terms: 1.5 + n = 2.872281323 Solving 1.5 + n = 2.872281323 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 = 2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.872281323 + -1.5 n = 2.872281323 + -1.5 Combine like terms: 2.872281323 + -1.5 = 1.372281323 n = 1.372281323 Simplifying n = 1.372281323Subproblem 2
n + 1.5 = -2.872281323 Simplifying n + 1.5 = -2.872281323 Reorder the terms: 1.5 + n = -2.872281323 Solving 1.5 + n = -2.872281323 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 = -2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.872281323 + -1.5 n = -2.872281323 + -1.5 Combine like terms: -2.872281323 + -1.5 = -4.372281323 n = -4.372281323 Simplifying n = -4.372281323Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.372281323, -4.372281323}
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