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Simplifying 3n2 + 17n = 4000 Reorder the terms: 17n + 3n2 = 4000 Solving 17n + 3n2 = 4000 Solving for variable 'n'. Reorder the terms: -4000 + 17n + 3n2 = 4000 + -4000 Combine like terms: 4000 + -4000 = 0 -4000 + 17n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1333.333333 + 5.666666667n + n2 = 0 Move the constant term to the right: Add '1333.333333' to each side of the equation. -1333.333333 + 5.666666667n + 1333.333333 + n2 = 0 + 1333.333333 Reorder the terms: -1333.333333 + 1333.333333 + 5.666666667n + n2 = 0 + 1333.333333 Combine like terms: -1333.333333 + 1333.333333 = 0.000000 0.000000 + 5.666666667n + n2 = 0 + 1333.333333 5.666666667n + n2 = 0 + 1333.333333 Combine like terms: 0 + 1333.333333 = 1333.333333 5.666666667n + n2 = 1333.333333 The n term is 5.666666667n. Take half its coefficient (2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. 5.666666667n + 8.027777782 + n2 = 1333.333333 + 8.027777782 Reorder the terms: 8.027777782 + 5.666666667n + n2 = 1333.333333 + 8.027777782 Combine like terms: 1333.333333 + 8.027777782 = 1341.361110782 8.027777782 + 5.666666667n + n2 = 1341.361110782 Factor a perfect square on the left side: (n + 2.833333334)(n + 2.833333334) = 1341.361110782 Calculate the square root of the right side: 36.624597073 Break this problem into two subproblems by setting (n + 2.833333334) equal to 36.624597073 and -36.624597073.Subproblem 1
n + 2.833333334 = 36.624597073 Simplifying n + 2.833333334 = 36.624597073 Reorder the terms: 2.833333334 + n = 36.624597073 Solving 2.833333334 + n = 36.624597073 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + n = 36.624597073 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + n = 36.624597073 + -2.833333334 n = 36.624597073 + -2.833333334 Combine like terms: 36.624597073 + -2.833333334 = 33.791263739 n = 33.791263739 Simplifying n = 33.791263739Subproblem 2
n + 2.833333334 = -36.624597073 Simplifying n + 2.833333334 = -36.624597073 Reorder the terms: 2.833333334 + n = -36.624597073 Solving 2.833333334 + n = -36.624597073 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + n = -36.624597073 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + n = -36.624597073 + -2.833333334 n = -36.624597073 + -2.833333334 Combine like terms: -36.624597073 + -2.833333334 = -39.457930407 n = -39.457930407 Simplifying n = -39.457930407Solution
The solution to the problem is based on the solutions from the subproblems. n = {33.791263739, -39.457930407}
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