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Simplifying 10n2 + -8n = 1 Reorder the terms: -8n + 10n2 = 1 Solving -8n + 10n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + -8n + 10n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -8n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.1 + -0.8n + n2 = 0 Move the constant term to the right: Add '0.1' to each side of the equation. -0.1 + -0.8n + 0.1 + n2 = 0 + 0.1 Reorder the terms: -0.1 + 0.1 + -0.8n + n2 = 0 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + -0.8n + n2 = 0 + 0.1 -0.8n + n2 = 0 + 0.1 Combine like terms: 0 + 0.1 = 0.1 -0.8n + n2 = 0.1 The n term is -0.8n. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8n + 0.16 + n2 = 0.1 + 0.16 Reorder the terms: 0.16 + -0.8n + n2 = 0.1 + 0.16 Combine like terms: 0.1 + 0.16 = 0.26 0.16 + -0.8n + n2 = 0.26 Factor a perfect square on the left side: (n + -0.4)(n + -0.4) = 0.26 Calculate the square root of the right side: 0.509901951 Break this problem into two subproblems by setting (n + -0.4) equal to 0.509901951 and -0.509901951.Subproblem 1
n + -0.4 = 0.509901951 Simplifying n + -0.4 = 0.509901951 Reorder the terms: -0.4 + n = 0.509901951 Solving -0.4 + n = 0.509901951 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = 0.509901951 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = 0.509901951 + 0.4 n = 0.509901951 + 0.4 Combine like terms: 0.509901951 + 0.4 = 0.909901951 n = 0.909901951 Simplifying n = 0.909901951Subproblem 2
n + -0.4 = -0.509901951 Simplifying n + -0.4 = -0.509901951 Reorder the terms: -0.4 + n = -0.509901951 Solving -0.4 + n = -0.509901951 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + n = -0.509901951 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = -0.509901951 + 0.4 n = -0.509901951 + 0.4 Combine like terms: -0.509901951 + 0.4 = -0.109901951 n = -0.109901951 Simplifying n = -0.109901951Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.909901951, -0.109901951}
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