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Simplifying 5n2 + 11n = 14 Reorder the terms: 11n + 5n2 = 14 Solving 11n + 5n2 = 14 Solving for variable 'n'. Reorder the terms: -14 + 11n + 5n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 11n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.8 + 2.2n + n2 = 0 Move the constant term to the right: Add '2.8' to each side of the equation. -2.8 + 2.2n + 2.8 + n2 = 0 + 2.8 Reorder the terms: -2.8 + 2.8 + 2.2n + n2 = 0 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + 2.2n + n2 = 0 + 2.8 2.2n + n2 = 0 + 2.8 Combine like terms: 0 + 2.8 = 2.8 2.2n + n2 = 2.8 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 = 2.8 + 1.21 Reorder the terms: 1.21 + 2.2n + n2 = 2.8 + 1.21 Combine like terms: 2.8 + 1.21 = 4.01 1.21 + 2.2n + n2 = 4.01 Factor a perfect square on the left side: (n + 1.1)(n + 1.1) = 4.01 Calculate the square root of the right side: 2.002498439 Break this problem into two subproblems by setting (n + 1.1) equal to 2.002498439 and -2.002498439.Subproblem 1
n + 1.1 = 2.002498439 Simplifying n + 1.1 = 2.002498439 Reorder the terms: 1.1 + n = 2.002498439 Solving 1.1 + n = 2.002498439 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 = 2.002498439 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = 2.002498439 + -1.1 n = 2.002498439 + -1.1 Combine like terms: 2.002498439 + -1.1 = 0.902498439 n = 0.902498439 Simplifying n = 0.902498439Subproblem 2
n + 1.1 = -2.002498439 Simplifying n + 1.1 = -2.002498439 Reorder the terms: 1.1 + n = -2.002498439 Solving 1.1 + n = -2.002498439 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 = -2.002498439 + -1.1 Combine like terms: 1.1 + -1.1 = 0.0 0.0 + n = -2.002498439 + -1.1 n = -2.002498439 + -1.1 Combine like terms: -2.002498439 + -1.1 = -3.102498439 n = -3.102498439 Simplifying n = -3.102498439Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.902498439, -3.102498439}
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