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Simplifying 2n(n) + 33n + -120 = 0 Multiply n * n 2n2 + 33n + -120 = 0 Reorder the terms: -120 + 33n + 2n2 = 0 Solving -120 + 33n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -60 + 16.5n + n2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 16.5n + 60 + n2 = 0 + 60 Reorder the terms: -60 + 60 + 16.5n + n2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 16.5n + n2 = 0 + 60 16.5n + n2 = 0 + 60 Combine like terms: 0 + 60 = 60 16.5n + n2 = 60 The n term is 16.5n. Take half its coefficient (8.25). Square it (68.0625) and add it to both sides. Add '68.0625' to each side of the equation. 16.5n + 68.0625 + n2 = 60 + 68.0625 Reorder the terms: 68.0625 + 16.5n + n2 = 60 + 68.0625 Combine like terms: 60 + 68.0625 = 128.0625 68.0625 + 16.5n + n2 = 128.0625 Factor a perfect square on the left side: (n + 8.25)(n + 8.25) = 128.0625 Calculate the square root of the right side: 11.316470298 Break this problem into two subproblems by setting (n + 8.25) equal to 11.316470298 and -11.316470298.Subproblem 1
n + 8.25 = 11.316470298 Simplifying n + 8.25 = 11.316470298 Reorder the terms: 8.25 + n = 11.316470298 Solving 8.25 + n = 11.316470298 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8.25' to each side of the equation. 8.25 + -8.25 + n = 11.316470298 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + n = 11.316470298 + -8.25 n = 11.316470298 + -8.25 Combine like terms: 11.316470298 + -8.25 = 3.066470298 n = 3.066470298 Simplifying n = 3.066470298Subproblem 2
n + 8.25 = -11.316470298 Simplifying n + 8.25 = -11.316470298 Reorder the terms: 8.25 + n = -11.316470298 Solving 8.25 + n = -11.316470298 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8.25' to each side of the equation. 8.25 + -8.25 + n = -11.316470298 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + n = -11.316470298 + -8.25 n = -11.316470298 + -8.25 Combine like terms: -11.316470298 + -8.25 = -19.566470298 n = -19.566470298 Simplifying n = -19.566470298Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.066470298, -19.566470298}
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