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Simplifying 8n(5n + 4) = 11 Reorder the terms: 8n(4 + 5n) = 11 (4 * 8n + 5n * 8n) = 11 (32n + 40n2) = 11 Solving 32n + 40n2 = 11 Solving for variable 'n'. Reorder the terms: -11 + 32n + 40n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 32n + 40n2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.275 + 0.8n + n2 = 0.0 Move the constant term to the right: Add '0.275' to each side of the equation. -0.275 + 0.8n + 0.275 + n2 = 0.0 + 0.275 Reorder the terms: -0.275 + 0.275 + 0.8n + n2 = 0.0 + 0.275 Combine like terms: -0.275 + 0.275 = 0.000 0.000 + 0.8n + n2 = 0.0 + 0.275 0.8n + n2 = 0.0 + 0.275 Combine like terms: 0.0 + 0.275 = 0.275 0.8n + n2 = 0.275 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 = 0.275 + 0.16 Reorder the terms: 0.16 + 0.8n + n2 = 0.275 + 0.16 Combine like terms: 0.275 + 0.16 = 0.435 0.16 + 0.8n + n2 = 0.435 Factor a perfect square on the left side: (n + 0.4)(n + 0.4) = 0.435 Calculate the square root of the right side: 0.659545298 Break this problem into two subproblems by setting (n + 0.4) equal to 0.659545298 and -0.659545298.Subproblem 1
n + 0.4 = 0.659545298 Simplifying n + 0.4 = 0.659545298 Reorder the terms: 0.4 + n = 0.659545298 Solving 0.4 + n = 0.659545298 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 = 0.659545298 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = 0.659545298 + -0.4 n = 0.659545298 + -0.4 Combine like terms: 0.659545298 + -0.4 = 0.259545298 n = 0.259545298 Simplifying n = 0.259545298Subproblem 2
n + 0.4 = -0.659545298 Simplifying n + 0.4 = -0.659545298 Reorder the terms: 0.4 + n = -0.659545298 Solving 0.4 + n = -0.659545298 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 = -0.659545298 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = -0.659545298 + -0.4 n = -0.659545298 + -0.4 Combine like terms: -0.659545298 + -0.4 = -1.059545298 n = -1.059545298 Simplifying n = -1.059545298Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.259545298, -1.059545298}
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