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Simplifying 11n2 + -5n = 12 Reorder the terms: -5n + 11n2 = 12 Solving -5n + 11n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + -5n + 11n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + -5n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.090909091 + -0.4545454545n + n2 = 0 Move the constant term to the right: Add '1.090909091' to each side of the equation. -1.090909091 + -0.4545454545n + 1.090909091 + n2 = 0 + 1.090909091 Reorder the terms: -1.090909091 + 1.090909091 + -0.4545454545n + n2 = 0 + 1.090909091 Combine like terms: -1.090909091 + 1.090909091 = 0.000000000 0.000000000 + -0.4545454545n + n2 = 0 + 1.090909091 -0.4545454545n + n2 = 0 + 1.090909091 Combine like terms: 0 + 1.090909091 = 1.090909091 -0.4545454545n + n2 = 1.090909091 The n term is -0.4545454545n. Take half its coefficient (-0.2272727273). Square it (0.05165289257) and add it to both sides. Add '0.05165289257' to each side of the equation. -0.4545454545n + 0.05165289257 + n2 = 1.090909091 + 0.05165289257 Reorder the terms: 0.05165289257 + -0.4545454545n + n2 = 1.090909091 + 0.05165289257 Combine like terms: 1.090909091 + 0.05165289257 = 1.14256198357 0.05165289257 + -0.4545454545n + n2 = 1.14256198357 Factor a perfect square on the left side: (n + -0.2272727273)(n + -0.2272727273) = 1.14256198357 Calculate the square root of the right side: 1.068906911 Break this problem into two subproblems by setting (n + -0.2272727273) equal to 1.068906911 and -1.068906911.Subproblem 1
n + -0.2272727273 = 1.068906911 Simplifying n + -0.2272727273 = 1.068906911 Reorder the terms: -0.2272727273 + n = 1.068906911 Solving -0.2272727273 + n = 1.068906911 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2272727273' to each side of the equation. -0.2272727273 + 0.2272727273 + n = 1.068906911 + 0.2272727273 Combine like terms: -0.2272727273 + 0.2272727273 = 0.0000000000 0.0000000000 + n = 1.068906911 + 0.2272727273 n = 1.068906911 + 0.2272727273 Combine like terms: 1.068906911 + 0.2272727273 = 1.2961796383 n = 1.2961796383 Simplifying n = 1.2961796383Subproblem 2
n + -0.2272727273 = -1.068906911 Simplifying n + -0.2272727273 = -1.068906911 Reorder the terms: -0.2272727273 + n = -1.068906911 Solving -0.2272727273 + n = -1.068906911 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2272727273' to each side of the equation. -0.2272727273 + 0.2272727273 + n = -1.068906911 + 0.2272727273 Combine like terms: -0.2272727273 + 0.2272727273 = 0.0000000000 0.0000000000 + n = -1.068906911 + 0.2272727273 n = -1.068906911 + 0.2272727273 Combine like terms: -1.068906911 + 0.2272727273 = -0.8416341837 n = -0.8416341837 Simplifying n = -0.8416341837Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.2961796383, -0.8416341837}
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