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Simplifying n2 + -3n + -31 = 3 Reorder the terms: -31 + -3n + n2 = 3 Solving -31 + -3n + n2 = 3 Solving for variable 'n'. Reorder the terms: -31 + -3 + -3n + n2 = 3 + -3 Combine like terms: -31 + -3 = -34 -34 + -3n + n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -34 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '34' to each side of the equation. -34 + -3n + 34 + n2 = 0 + 34 Reorder the terms: -34 + 34 + -3n + n2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + -3n + n2 = 0 + 34 -3n + n2 = 0 + 34 Combine like terms: 0 + 34 = 34 -3n + n2 = 34 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 = 34 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 34 + 2.25 Combine like terms: 34 + 2.25 = 36.25 2.25 + -3n + n2 = 36.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 36.25 Calculate the square root of the right side: 6.020797289 Break this problem into two subproblems by setting (n + -1.5) equal to 6.020797289 and -6.020797289.Subproblem 1
n + -1.5 = 6.020797289 Simplifying n + -1.5 = 6.020797289 Reorder the terms: -1.5 + n = 6.020797289 Solving -1.5 + n = 6.020797289 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 = 6.020797289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.020797289 + 1.5 n = 6.020797289 + 1.5 Combine like terms: 6.020797289 + 1.5 = 7.520797289 n = 7.520797289 Simplifying n = 7.520797289Subproblem 2
n + -1.5 = -6.020797289 Simplifying n + -1.5 = -6.020797289 Reorder the terms: -1.5 + n = -6.020797289 Solving -1.5 + n = -6.020797289 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 = -6.020797289 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.020797289 + 1.5 n = -6.020797289 + 1.5 Combine like terms: -6.020797289 + 1.5 = -4.520797289 n = -4.520797289 Simplifying n = -4.520797289Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.520797289, -4.520797289}
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