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Simplifying n2 + 2n + -27 = -6 Reorder the terms: -27 + 2n + n2 = -6 Solving -27 + 2n + n2 = -6 Solving for variable 'n'. Reorder the terms: -27 + 6 + 2n + n2 = -6 + 6 Combine like terms: -27 + 6 = -21 -21 + 2n + n2 = -6 + 6 Combine like terms: -6 + 6 = 0 -21 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + 2n + 21 + n2 = 0 + 21 Reorder the terms: -21 + 21 + 2n + n2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 2n + n2 = 0 + 21 2n + n2 = 0 + 21 Combine like terms: 0 + 21 = 21 2n + n2 = 21 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 21 + 1 Reorder the terms: 1 + 2n + n2 = 21 + 1 Combine like terms: 21 + 1 = 22 1 + 2n + n2 = 22 Factor a perfect square on the left side: (n + 1)(n + 1) = 22 Calculate the square root of the right side: 4.69041576 Break this problem into two subproblems by setting (n + 1) equal to 4.69041576 and -4.69041576.Subproblem 1
n + 1 = 4.69041576 Simplifying n + 1 = 4.69041576 Reorder the terms: 1 + n = 4.69041576 Solving 1 + n = 4.69041576 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + n = 4.69041576 + -1 n = 4.69041576 + -1 Combine like terms: 4.69041576 + -1 = 3.69041576 n = 3.69041576 Simplifying n = 3.69041576Subproblem 2
n + 1 = -4.69041576 Simplifying n + 1 = -4.69041576 Reorder the terms: 1 + n = -4.69041576 Solving 1 + n = -4.69041576 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + n = -4.69041576 + -1 n = -4.69041576 + -1 Combine like terms: -4.69041576 + -1 = -5.69041576 n = -5.69041576 Simplifying n = -5.69041576Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.69041576, -5.69041576}
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