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Simplifying 15n2 + 112n + 41 = 0 Reorder the terms: 41 + 112n + 15n2 = 0 Solving 41 + 112n + 15n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 2.733333333 + 7.466666667n + n2 = 0 Move the constant term to the right: Add '-2.733333333' to each side of the equation. 2.733333333 + 7.466666667n + -2.733333333 + n2 = 0 + -2.733333333 Reorder the terms: 2.733333333 + -2.733333333 + 7.466666667n + n2 = 0 + -2.733333333 Combine like terms: 2.733333333 + -2.733333333 = 0.000000000 0.000000000 + 7.466666667n + n2 = 0 + -2.733333333 7.466666667n + n2 = 0 + -2.733333333 Combine like terms: 0 + -2.733333333 = -2.733333333 7.466666667n + n2 = -2.733333333 The n term is 7.466666667n. Take half its coefficient (3.733333334). Square it (13.93777778) and add it to both sides. Add '13.93777778' to each side of the equation. 7.466666667n + 13.93777778 + n2 = -2.733333333 + 13.93777778 Reorder the terms: 13.93777778 + 7.466666667n + n2 = -2.733333333 + 13.93777778 Combine like terms: -2.733333333 + 13.93777778 = 11.204444447 13.93777778 + 7.466666667n + n2 = 11.204444447 Factor a perfect square on the left side: (n + 3.733333334)(n + 3.733333334) = 11.204444447 Calculate the square root of the right side: 3.347304057 Break this problem into two subproblems by setting (n + 3.733333334) equal to 3.347304057 and -3.347304057.Subproblem 1
n + 3.733333334 = 3.347304057 Simplifying n + 3.733333334 = 3.347304057 Reorder the terms: 3.733333334 + n = 3.347304057 Solving 3.733333334 + n = 3.347304057 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.733333334' to each side of the equation. 3.733333334 + -3.733333334 + n = 3.347304057 + -3.733333334 Combine like terms: 3.733333334 + -3.733333334 = 0.000000000 0.000000000 + n = 3.347304057 + -3.733333334 n = 3.347304057 + -3.733333334 Combine like terms: 3.347304057 + -3.733333334 = -0.386029277 n = -0.386029277 Simplifying n = -0.386029277Subproblem 2
n + 3.733333334 = -3.347304057 Simplifying n + 3.733333334 = -3.347304057 Reorder the terms: 3.733333334 + n = -3.347304057 Solving 3.733333334 + n = -3.347304057 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.733333334' to each side of the equation. 3.733333334 + -3.733333334 + n = -3.347304057 + -3.733333334 Combine like terms: 3.733333334 + -3.733333334 = 0.000000000 0.000000000 + n = -3.347304057 + -3.733333334 n = -3.347304057 + -3.733333334 Combine like terms: -3.347304057 + -3.733333334 = -7.080637391 n = -7.080637391 Simplifying n = -7.080637391Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.386029277, -7.080637391}
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