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Simplifying n[2.2 + 0.2n] + -2035.2 = 0 [2.2 * n + 0.2n * n] + -2035.2 = 0 [2.2n + 0.2n2] + -2035.2 = 0 Reorder the terms: -2035.2 + 2.2n + 0.2n2 = 0 Solving -2035.2 + 2.2n + 0.2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. -10176 + 11n + n2 = 0 Move the constant term to the right: Add '10176' to each side of the equation. -10176 + 11n + 10176 + n2 = 0 + 10176 Reorder the terms: -10176 + 10176 + 11n + n2 = 0 + 10176 Combine like terms: -10176 + 10176 = 0 0 + 11n + n2 = 0 + 10176 11n + n2 = 0 + 10176 Combine like terms: 0 + 10176 = 10176 11n + n2 = 10176 The n term is 11n. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11n + 30.25 + n2 = 10176 + 30.25 Reorder the terms: 30.25 + 11n + n2 = 10176 + 30.25 Combine like terms: 10176 + 30.25 = 10206.25 30.25 + 11n + n2 = 10206.25 Factor a perfect square on the left side: (n + 5.5)(n + 5.5) = 10206.25 Calculate the square root of the right side: 101.025986756 Break this problem into two subproblems by setting (n + 5.5) equal to 101.025986756 and -101.025986756.Subproblem 1
n + 5.5 = 101.025986756 Simplifying n + 5.5 = 101.025986756 Reorder the terms: 5.5 + n = 101.025986756 Solving 5.5 + n = 101.025986756 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = 101.025986756 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = 101.025986756 + -5.5 n = 101.025986756 + -5.5 Combine like terms: 101.025986756 + -5.5 = 95.525986756 n = 95.525986756 Simplifying n = 95.525986756Subproblem 2
n + 5.5 = -101.025986756 Simplifying n + 5.5 = -101.025986756 Reorder the terms: 5.5 + n = -101.025986756 Solving 5.5 + n = -101.025986756 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = -101.025986756 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = -101.025986756 + -5.5 n = -101.025986756 + -5.5 Combine like terms: -101.025986756 + -5.5 = -106.525986756 n = -106.525986756 Simplifying n = -106.525986756Solution
The solution to the problem is based on the solutions from the subproblems. n = {95.525986756, -106.525986756}
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