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Simplifying 5n2 + 11n = 2000 Reorder the terms: 11n + 5n2 = 2000 Solving 11n + 5n2 = 2000 Solving for variable 'n'. Reorder the terms: -2000 + 11n + 5n2 = 2000 + -2000 Combine like terms: 2000 + -2000 = 0 -2000 + 11n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -400 + 2.2n + n2 = 0 Move the constant term to the right: Add '400' to each side of the equation. -400 + 2.2n + 400 + n2 = 0 + 400 Reorder the terms: -400 + 400 + 2.2n + n2 = 0 + 400 Combine like terms: -400 + 400 = 0 0 + 2.2n + n2 = 0 + 400 2.2n + n2 = 0 + 400 Combine like terms: 0 + 400 = 400 2.2n + n2 = 400 The n term is 2.2n. Take half its coefficient (1.1). Square it (1.21) and add it to both sides. Add '1.21' to each side of the equation. 2.2n + 1.21 + n2 = 400 + 1.21 Reorder the terms: 1.21 + 2.2n + n2 = 400 + 1.21 Combine like terms: 400 + 1.21 = 401.21 1.21 + 2.2n + n2 = 401.21 Factor a perfect square on the left side: (n + 1.1)(n + 1.1) = 401.21 Calculate the square root of the right side: 20.030227158 Break this problem into two subproblems by setting (n + 1.1) equal to 20.030227158 and -20.030227158.Subproblem 1
n + 1.1 = 20.030227158 Simplifying n + 1.1 = 20.030227158 Reorder the terms: 1.1 + n = 20.030227158 Solving 1.1 + n = 20.030227158 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.1' to each side of the equation. 1.1 + -1.1 + n = 20.030227158 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = 20.030227158 + -1.1 n = 20.030227158 + -1.1 Combine like terms: 20.030227158 + -1.1 = 18.930227158 n = 18.930227158 Simplifying n = 18.930227158Subproblem 2
n + 1.1 = -20.030227158 Simplifying n + 1.1 = -20.030227158 Reorder the terms: 1.1 + n = -20.030227158 Solving 1.1 + n = -20.030227158 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.1' to each side of the equation. 1.1 + -1.1 + n = -20.030227158 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = -20.030227158 + -1.1 n = -20.030227158 + -1.1 Combine like terms: -20.030227158 + -1.1 = -21.130227158 n = -21.130227158 Simplifying n = -21.130227158Solution
The solution to the problem is based on the solutions from the subproblems. n = {18.930227158, -21.130227158}
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