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Simplifying 11n2 + -1 = 9n Reorder the terms: -1 + 11n2 = 9n Solving -1 + 11n2 = 9n Solving for variable 'n'. Reorder the terms: -1 + -9n + 11n2 = 9n + -9n Combine like terms: 9n + -9n = 0 -1 + -9n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.09090909091 + -0.8181818182n + n2 = 0 Move the constant term to the right: Add '0.09090909091' to each side of the equation. -0.09090909091 + -0.8181818182n + 0.09090909091 + n2 = 0 + 0.09090909091 Reorder the terms: -0.09090909091 + 0.09090909091 + -0.8181818182n + n2 = 0 + 0.09090909091 Combine like terms: -0.09090909091 + 0.09090909091 = 0.00000000000 0.00000000000 + -0.8181818182n + n2 = 0 + 0.09090909091 -0.8181818182n + n2 = 0 + 0.09090909091 Combine like terms: 0 + 0.09090909091 = 0.09090909091 -0.8181818182n + n2 = 0.09090909091 The n term is -0.8181818182n. Take half its coefficient (-0.4090909091). Square it (0.1673553719) and add it to both sides. Add '0.1673553719' to each side of the equation. -0.8181818182n + 0.1673553719 + n2 = 0.09090909091 + 0.1673553719 Reorder the terms: 0.1673553719 + -0.8181818182n + n2 = 0.09090909091 + 0.1673553719 Combine like terms: 0.09090909091 + 0.1673553719 = 0.25826446281 0.1673553719 + -0.8181818182n + n2 = 0.25826446281 Factor a perfect square on the left side: (n + -0.4090909091)(n + -0.4090909091) = 0.25826446281 Calculate the square root of the right side: 0.508197268 Break this problem into two subproblems by setting (n + -0.4090909091) equal to 0.508197268 and -0.508197268.Subproblem 1
n + -0.4090909091 = 0.508197268 Simplifying n + -0.4090909091 = 0.508197268 Reorder the terms: -0.4090909091 + n = 0.508197268 Solving -0.4090909091 + n = 0.508197268 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4090909091' to each side of the equation. -0.4090909091 + 0.4090909091 + n = 0.508197268 + 0.4090909091 Combine like terms: -0.4090909091 + 0.4090909091 = 0.0000000000 0.0000000000 + n = 0.508197268 + 0.4090909091 n = 0.508197268 + 0.4090909091 Combine like terms: 0.508197268 + 0.4090909091 = 0.9172881771 n = 0.9172881771 Simplifying n = 0.9172881771Subproblem 2
n + -0.4090909091 = -0.508197268 Simplifying n + -0.4090909091 = -0.508197268 Reorder the terms: -0.4090909091 + n = -0.508197268 Solving -0.4090909091 + n = -0.508197268 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4090909091' to each side of the equation. -0.4090909091 + 0.4090909091 + n = -0.508197268 + 0.4090909091 Combine like terms: -0.4090909091 + 0.4090909091 = 0.0000000000 0.0000000000 + n = -0.508197268 + 0.4090909091 n = -0.508197268 + 0.4090909091 Combine like terms: -0.508197268 + 0.4090909091 = -0.0991063589 n = -0.0991063589 Simplifying n = -0.0991063589Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.9172881771, -0.0991063589}
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