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Simplifying 6n = n2 + 2 Reorder the terms: 6n = 2 + n2 Solving 6n = 2 + n2 Solving for variable 'n'. Reorder the terms: -2 + 6n + -1n2 = 2 + n2 + -2 + -1n2 Reorder the terms: -2 + 6n + -1n2 = 2 + -2 + n2 + -1n2 Combine like terms: 2 + -2 = 0 -2 + 6n + -1n2 = 0 + n2 + -1n2 -2 + 6n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 -2 + 6n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 2 + -6n + n2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -6n + -2 + n2 = 0 + -2 Reorder the terms: 2 + -2 + -6n + n2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -6n + n2 = 0 + -2 -6n + n2 = 0 + -2 Combine like terms: 0 + -2 = -2 -6n + n2 = -2 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -2 + 9 Reorder the terms: 9 + -6n + n2 = -2 + 9 Combine like terms: -2 + 9 = 7 9 + -6n + n2 = 7 Factor a perfect square on the left side: (n + -3)(n + -3) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (n + -3) equal to 2.645751311 and -2.645751311.Subproblem 1
n + -3 = 2.645751311 Simplifying n + -3 = 2.645751311 Reorder the terms: -3 + n = 2.645751311 Solving -3 + n = 2.645751311 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.645751311 + 3 n = 2.645751311 + 3 Combine like terms: 2.645751311 + 3 = 5.645751311 n = 5.645751311 Simplifying n = 5.645751311Subproblem 2
n + -3 = -2.645751311 Simplifying n + -3 = -2.645751311 Reorder the terms: -3 + n = -2.645751311 Solving -3 + n = -2.645751311 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.645751311 + 3 n = -2.645751311 + 3 Combine like terms: -2.645751311 + 3 = 0.354248689 n = 0.354248689 Simplifying n = 0.354248689Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.645751311, 0.354248689}
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