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Simplifying 270 = 3.7n2 + 8n + -3 Reorder the terms: 270 = -3 + 8n + 3.7n2 Solving 270 = -3 + 8n + 3.7n2 Solving for variable 'n'. Combine like terms: 270 + 3 = 273 273 + -8n + -3.7n2 = -3 + 8n + 3.7n2 + 3 + -8n + -3.7n2 Reorder the terms: 273 + -8n + -3.7n2 = -3 + 3 + 8n + -8n + 3.7n2 + -3.7n2 Combine like terms: -3 + 3 = 0 273 + -8n + -3.7n2 = 0 + 8n + -8n + 3.7n2 + -3.7n2 273 + -8n + -3.7n2 = 8n + -8n + 3.7n2 + -3.7n2 Combine like terms: 8n + -8n = 0 273 + -8n + -3.7n2 = 0 + 3.7n2 + -3.7n2 273 + -8n + -3.7n2 = 3.7n2 + -3.7n2 Combine like terms: 3.7n2 + -3.7n2 = 0.0 273 + -8n + -3.7n2 = 0.0 Begin completing the square. Divide all terms by -3.7 the coefficient of the squared term: Divide each side by '-3.7'. -73.78378378 + 2.162162162n + n2 = 0 Move the constant term to the right: Add '73.78378378' to each side of the equation. -73.78378378 + 2.162162162n + 73.78378378 + n2 = 0 + 73.78378378 Reorder the terms: -73.78378378 + 73.78378378 + 2.162162162n + n2 = 0 + 73.78378378 Combine like terms: -73.78378378 + 73.78378378 = 0.00000000 0.00000000 + 2.162162162n + n2 = 0 + 73.78378378 2.162162162n + n2 = 0 + 73.78378378 Combine like terms: 0 + 73.78378378 = 73.78378378 2.162162162n + n2 = 73.78378378 The n term is 2.162162162n. Take half its coefficient (1.081081081). Square it (1.168736304) and add it to both sides. Add '1.168736304' to each side of the equation. 2.162162162n + 1.168736304 + n2 = 73.78378378 + 1.168736304 Reorder the terms: 1.168736304 + 2.162162162n + n2 = 73.78378378 + 1.168736304 Combine like terms: 73.78378378 + 1.168736304 = 74.952520084 1.168736304 + 2.162162162n + n2 = 74.952520084 Factor a perfect square on the left side: (n + 1.081081081)(n + 1.081081081) = 74.952520084 Calculate the square root of the right side: 8.65751235 Break this problem into two subproblems by setting (n + 1.081081081) equal to 8.65751235 and -8.65751235.Subproblem 1
n + 1.081081081 = 8.65751235 Simplifying n + 1.081081081 = 8.65751235 Reorder the terms: 1.081081081 + n = 8.65751235 Solving 1.081081081 + n = 8.65751235 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.081081081' to each side of the equation. 1.081081081 + -1.081081081 + n = 8.65751235 + -1.081081081 Combine like terms: 1.081081081 + -1.081081081 = 0.000000000 0.000000000 + n = 8.65751235 + -1.081081081 n = 8.65751235 + -1.081081081 Combine like terms: 8.65751235 + -1.081081081 = 7.576431269 n = 7.576431269 Simplifying n = 7.576431269Subproblem 2
n + 1.081081081 = -8.65751235 Simplifying n + 1.081081081 = -8.65751235 Reorder the terms: 1.081081081 + n = -8.65751235 Solving 1.081081081 + n = -8.65751235 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.081081081' to each side of the equation. 1.081081081 + -1.081081081 + n = -8.65751235 + -1.081081081 Combine like terms: 1.081081081 + -1.081081081 = 0.000000000 0.000000000 + n = -8.65751235 + -1.081081081 n = -8.65751235 + -1.081081081 Combine like terms: -8.65751235 + -1.081081081 = -9.738593431 n = -9.738593431 Simplifying n = -9.738593431Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.576431269, -9.738593431}
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