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Simplifying 5n2 + 1 = -8n Reorder the terms: 1 + 5n2 = -8n Solving 1 + 5n2 = -8n Solving for variable 'n'. Reorder the terms: 1 + 8n + 5n2 = -8n + 8n Combine like terms: -8n + 8n = 0 1 + 8n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + 1.6n + n2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + 1.6n + -0.2 + n2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + 1.6n + n2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + 1.6n + n2 = 0 + -0.2 1.6n + n2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 1.6n + n2 = -0.2 The n term is 1.6n. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6n + 0.64 + n2 = -0.2 + 0.64 Reorder the terms: 0.64 + 1.6n + n2 = -0.2 + 0.64 Combine like terms: -0.2 + 0.64 = 0.44 0.64 + 1.6n + n2 = 0.44 Factor a perfect square on the left side: (n + 0.8)(n + 0.8) = 0.44 Calculate the square root of the right side: 0.663324958 Break this problem into two subproblems by setting (n + 0.8) equal to 0.663324958 and -0.663324958.Subproblem 1
n + 0.8 = 0.663324958 Simplifying n + 0.8 = 0.663324958 Reorder the terms: 0.8 + n = 0.663324958 Solving 0.8 + n = 0.663324958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + n = 0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = 0.663324958 + -0.8 n = 0.663324958 + -0.8 Combine like terms: 0.663324958 + -0.8 = -0.136675042 n = -0.136675042 Simplifying n = -0.136675042Subproblem 2
n + 0.8 = -0.663324958 Simplifying n + 0.8 = -0.663324958 Reorder the terms: 0.8 + n = -0.663324958 Solving 0.8 + n = -0.663324958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + n = -0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + n = -0.663324958 + -0.8 n = -0.663324958 + -0.8 Combine like terms: -0.663324958 + -0.8 = -1.463324958 n = -1.463324958 Simplifying n = -1.463324958Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.136675042, -1.463324958}
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