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Simplifying -11n2 + 8n + 22 = 12 Reorder the terms: 22 + 8n + -11n2 = 12 Solving 22 + 8n + -11n2 = 12 Solving for variable 'n'. Reorder the terms: 22 + -12 + 8n + -11n2 = 12 + -12 Combine like terms: 22 + -12 = 10 10 + 8n + -11n2 = 12 + -12 Combine like terms: 12 + -12 = 0 10 + 8n + -11n2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -0.9090909091 + -0.7272727273n + n2 = 0 Move the constant term to the right: Add '0.9090909091' to each side of the equation. -0.9090909091 + -0.7272727273n + 0.9090909091 + n2 = 0 + 0.9090909091 Reorder the terms: -0.9090909091 + 0.9090909091 + -0.7272727273n + n2 = 0 + 0.9090909091 Combine like terms: -0.9090909091 + 0.9090909091 = 0.0000000000 0.0000000000 + -0.7272727273n + n2 = 0 + 0.9090909091 -0.7272727273n + n2 = 0 + 0.9090909091 Combine like terms: 0 + 0.9090909091 = 0.9090909091 -0.7272727273n + n2 = 0.9090909091 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 = 0.9090909091 + 0.1322314050 Reorder the terms: 0.1322314050 + -0.7272727273n + n2 = 0.9090909091 + 0.1322314050 Combine like terms: 0.9090909091 + 0.1322314050 = 1.0413223141 0.1322314050 + -0.7272727273n + n2 = 1.0413223141 Factor a perfect square on the left side: (n + -0.3636363637)(n + -0.3636363637) = 1.0413223141 Calculate the square root of the right side: 1.020452015 Break this problem into two subproblems by setting (n + -0.3636363637) equal to 1.020452015 and -1.020452015.Subproblem 1
n + -0.3636363637 = 1.020452015 Simplifying n + -0.3636363637 = 1.020452015 Reorder the terms: -0.3636363637 + n = 1.020452015 Solving -0.3636363637 + n = 1.020452015 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 = 1.020452015 + 0.3636363637 Combine like terms: -0.3636363637 + 0.3636363637 = 0.0000000000 0.0000000000 + n = 1.020452015 + 0.3636363637 n = 1.020452015 + 0.3636363637 Combine like terms: 1.020452015 + 0.3636363637 = 1.3840883787 n = 1.3840883787 Simplifying n = 1.3840883787Subproblem 2
n + -0.3636363637 = -1.020452015 Simplifying n + -0.3636363637 = -1.020452015 Reorder the terms: -0.3636363637 + n = -1.020452015 Solving -0.3636363637 + n = -1.020452015 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 = -1.020452015 + 0.3636363637 Combine like terms: -0.3636363637 + 0.3636363637 = 0.0000000000 0.0000000000 + n = -1.020452015 + 0.3636363637 n = -1.020452015 + 0.3636363637 Combine like terms: -1.020452015 + 0.3636363637 = -0.6568156513 n = -0.6568156513 Simplifying n = -0.6568156513Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.3840883787, -0.6568156513}
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