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Simplifying 10n2 + -3 = 8n Reorder the terms: -3 + 10n2 = 8n Solving -3 + 10n2 = 8n Solving for variable 'n'. Reorder the terms: -3 + -8n + 10n2 = 8n + -8n Combine like terms: 8n + -8n = 0 -3 + -8n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.3 + -0.8n + n2 = 0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + -0.8n + 0.3 + n2 = 0 + 0.3 Reorder the terms: -0.3 + 0.3 + -0.8n + n2 = 0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + -0.8n + n2 = 0 + 0.3 -0.8n + n2 = 0 + 0.3 Combine like terms: 0 + 0.3 = 0.3 -0.8n + n2 = 0.3 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 = 0.3 + 0.16 Reorder the terms: 0.16 + -0.8n + n2 = 0.3 + 0.16 Combine like terms: 0.3 + 0.16 = 0.46 0.16 + -0.8n + n2 = 0.46 Factor a perfect square on the left side: (n + -0.4)(n + -0.4) = 0.46 Calculate the square root of the right side: 0.678232998 Break this problem into two subproblems by setting (n + -0.4) equal to 0.678232998 and -0.678232998.Subproblem 1
n + -0.4 = 0.678232998 Simplifying n + -0.4 = 0.678232998 Reorder the terms: -0.4 + n = 0.678232998 Solving -0.4 + n = 0.678232998 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 = 0.678232998 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = 0.678232998 + 0.4 n = 0.678232998 + 0.4 Combine like terms: 0.678232998 + 0.4 = 1.078232998 n = 1.078232998 Simplifying n = 1.078232998Subproblem 2
n + -0.4 = -0.678232998 Simplifying n + -0.4 = -0.678232998 Reorder the terms: -0.4 + n = -0.678232998 Solving -0.4 + n = -0.678232998 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 = -0.678232998 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + n = -0.678232998 + 0.4 n = -0.678232998 + 0.4 Combine like terms: -0.678232998 + 0.4 = -0.278232998 n = -0.278232998 Simplifying n = -0.278232998Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.078232998, -0.278232998}
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