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Simplifying 2n2 + 8n + 1 = 0 Reorder the terms: 1 + 8n + 2n2 = 0 Solving 1 + 8n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 4n + n2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 4n + -0.5 + n2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 4n + n2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 4n + n2 = 0 + -0.5 4n + n2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 4n + n2 = -0.5 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = -0.5 + 4 Reorder the terms: 4 + 4n + n2 = -0.5 + 4 Combine like terms: -0.5 + 4 = 3.5 4 + 4n + n2 = 3.5 Factor a perfect square on the left side: (n + 2)(n + 2) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (n + 2) equal to 1.870828693 and -1.870828693.Subproblem 1
n + 2 = 1.870828693 Simplifying n + 2 = 1.870828693 Reorder the terms: 2 + n = 1.870828693 Solving 2 + n = 1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 1.870828693 + -2 Combine like terms: 2 + -2 = 0 0 + n = 1.870828693 + -2 n = 1.870828693 + -2 Combine like terms: 1.870828693 + -2 = -0.129171307 n = -0.129171307 Simplifying n = -0.129171307Subproblem 2
n + 2 = -1.870828693 Simplifying n + 2 = -1.870828693 Reorder the terms: 2 + n = -1.870828693 Solving 2 + n = -1.870828693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -1.870828693 + -2 Combine like terms: 2 + -2 = 0 0 + n = -1.870828693 + -2 n = -1.870828693 + -2 Combine like terms: -1.870828693 + -2 = -3.870828693 n = -3.870828693 Simplifying n = -3.870828693Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.129171307, -3.870828693}
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