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Simplifying 11n2 = 21 + -8n Solving 11n2 = 21 + -8n Solving for variable 'n'. Reorder the terms: -21 + 8n + 11n2 = 21 + -8n + -21 + 8n Reorder the terms: -21 + 8n + 11n2 = 21 + -21 + -8n + 8n Combine like terms: 21 + -21 = 0 -21 + 8n + 11n2 = 0 + -8n + 8n -21 + 8n + 11n2 = -8n + 8n Combine like terms: -8n + 8n = 0 -21 + 8n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.909090909 + 0.7272727273n + n2 = 0 Move the constant term to the right: Add '1.909090909' to each side of the equation. -1.909090909 + 0.7272727273n + 1.909090909 + n2 = 0 + 1.909090909 Reorder the terms: -1.909090909 + 1.909090909 + 0.7272727273n + n2 = 0 + 1.909090909 Combine like terms: -1.909090909 + 1.909090909 = 0.000000000 0.000000000 + 0.7272727273n + n2 = 0 + 1.909090909 0.7272727273n + n2 = 0 + 1.909090909 Combine like terms: 0 + 1.909090909 = 1.909090909 0.7272727273n + n2 = 1.909090909 The n term is 0.7272727273n. Take half its coefficient (0.3636363637). Square it (0.1322314050) and add it to both sides. Add '0.1322314050' to each side of the equation. 0.7272727273n + 0.1322314050 + n2 = 1.909090909 + 0.1322314050 Reorder the terms: 0.1322314050 + 0.7272727273n + n2 = 1.909090909 + 0.1322314050 Combine like terms: 1.909090909 + 0.1322314050 = 2.041322314 0.1322314050 + 0.7272727273n + n2 = 2.041322314 Factor a perfect square on the left side: (n + 0.3636363637)(n + 0.3636363637) = 2.041322314 Calculate the square root of the right side: 1.428748513 Break this problem into two subproblems by setting (n + 0.3636363637) equal to 1.428748513 and -1.428748513.Subproblem 1
n + 0.3636363637 = 1.428748513 Simplifying n + 0.3636363637 = 1.428748513 Reorder the terms: 0.3636363637 + n = 1.428748513 Solving 0.3636363637 + n = 1.428748513 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3636363637' to each side of the equation. 0.3636363637 + -0.3636363637 + n = 1.428748513 + -0.3636363637 Combine like terms: 0.3636363637 + -0.3636363637 = 0.0000000000 0.0000000000 + n = 1.428748513 + -0.3636363637 n = 1.428748513 + -0.3636363637 Combine like terms: 1.428748513 + -0.3636363637 = 1.0651121493 n = 1.0651121493 Simplifying n = 1.0651121493Subproblem 2
n + 0.3636363637 = -1.428748513 Simplifying n + 0.3636363637 = -1.428748513 Reorder the terms: 0.3636363637 + n = -1.428748513 Solving 0.3636363637 + n = -1.428748513 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3636363637' to each side of the equation. 0.3636363637 + -0.3636363637 + n = -1.428748513 + -0.3636363637 Combine like terms: 0.3636363637 + -0.3636363637 = 0.0000000000 0.0000000000 + n = -1.428748513 + -0.3636363637 n = -1.428748513 + -0.3636363637 Combine like terms: -1.428748513 + -0.3636363637 = -1.7923848767 n = -1.7923848767 Simplifying n = -1.7923848767Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.0651121493, -1.7923848767}
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