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Simplifying 3n2 + 77n + -6500 = 0 Reorder the terms: -6500 + 77n + 3n2 = 0 Solving -6500 + 77n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2166.666667 + 25.66666667n + n2 = 0 Move the constant term to the right: Add '2166.666667' to each side of the equation. -2166.666667 + 25.66666667n + 2166.666667 + n2 = 0 + 2166.666667 Reorder the terms: -2166.666667 + 2166.666667 + 25.66666667n + n2 = 0 + 2166.666667 Combine like terms: -2166.666667 + 2166.666667 = 0.000000 0.000000 + 25.66666667n + n2 = 0 + 2166.666667 25.66666667n + n2 = 0 + 2166.666667 Combine like terms: 0 + 2166.666667 = 2166.666667 25.66666667n + n2 = 2166.666667 The n term is 25.66666667n. Take half its coefficient (12.83333334). Square it (164.6944446) and add it to both sides. Add '164.6944446' to each side of the equation. 25.66666667n + 164.6944446 + n2 = 2166.666667 + 164.6944446 Reorder the terms: 164.6944446 + 25.66666667n + n2 = 2166.666667 + 164.6944446 Combine like terms: 2166.666667 + 164.6944446 = 2331.3611116 164.6944446 + 25.66666667n + n2 = 2331.3611116 Factor a perfect square on the left side: (n + 12.83333334)(n + 12.83333334) = 2331.3611116 Calculate the square root of the right side: 48.284170404 Break this problem into two subproblems by setting (n + 12.83333334) equal to 48.284170404 and -48.284170404.Subproblem 1
n + 12.83333334 = 48.284170404 Simplifying n + 12.83333334 = 48.284170404 Reorder the terms: 12.83333334 + n = 48.284170404 Solving 12.83333334 + n = 48.284170404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + n = 48.284170404 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + n = 48.284170404 + -12.83333334 n = 48.284170404 + -12.83333334 Combine like terms: 48.284170404 + -12.83333334 = 35.450837064 n = 35.450837064 Simplifying n = 35.450837064Subproblem 2
n + 12.83333334 = -48.284170404 Simplifying n + 12.83333334 = -48.284170404 Reorder the terms: 12.83333334 + n = -48.284170404 Solving 12.83333334 + n = -48.284170404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.83333334' to each side of the equation. 12.83333334 + -12.83333334 + n = -48.284170404 + -12.83333334 Combine like terms: 12.83333334 + -12.83333334 = 0.00000000 0.00000000 + n = -48.284170404 + -12.83333334 n = -48.284170404 + -12.83333334 Combine like terms: -48.284170404 + -12.83333334 = -61.117503744 n = -61.117503744 Simplifying n = -61.117503744Solution
The solution to the problem is based on the solutions from the subproblems. n = {35.450837064, -61.117503744}
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