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Simplifying 7n2 + 14n + 5 = 0 Reorder the terms: 5 + 14n + 7n2 = 0 Solving 5 + 14n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.7142857143 + 2n + n2 = 0 Move the constant term to the right: Add '-0.7142857143' to each side of the equation. 0.7142857143 + 2n + -0.7142857143 + n2 = 0 + -0.7142857143 Reorder the terms: 0.7142857143 + -0.7142857143 + 2n + n2 = 0 + -0.7142857143 Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000 0.0000000000 + 2n + n2 = 0 + -0.7142857143 2n + n2 = 0 + -0.7142857143 Combine like terms: 0 + -0.7142857143 = -0.7142857143 2n + n2 = -0.7142857143 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = -0.7142857143 + 1 Reorder the terms: 1 + 2n + n2 = -0.7142857143 + 1 Combine like terms: -0.7142857143 + 1 = 0.2857142857 1 + 2n + n2 = 0.2857142857 Factor a perfect square on the left side: (n + 1)(n + 1) = 0.2857142857 Calculate the square root of the right side: 0.534522484 Break this problem into two subproblems by setting (n + 1) equal to 0.534522484 and -0.534522484.Subproblem 1
n + 1 = 0.534522484 Simplifying n + 1 = 0.534522484 Reorder the terms: 1 + n = 0.534522484 Solving 1 + n = 0.534522484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 0.534522484 + -1 Combine like terms: 1 + -1 = 0 0 + n = 0.534522484 + -1 n = 0.534522484 + -1 Combine like terms: 0.534522484 + -1 = -0.465477516 n = -0.465477516 Simplifying n = -0.465477516Subproblem 2
n + 1 = -0.534522484 Simplifying n + 1 = -0.534522484 Reorder the terms: 1 + n = -0.534522484 Solving 1 + n = -0.534522484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -0.534522484 + -1 Combine like terms: 1 + -1 = 0 0 + n = -0.534522484 + -1 n = -0.534522484 + -1 Combine like terms: -0.534522484 + -1 = -1.534522484 n = -1.534522484 Simplifying n = -1.534522484Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.465477516, -1.534522484}
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