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Simplifying 12n = n2 + 31 Reorder the terms: 12n = 31 + n2 Solving 12n = 31 + n2 Solving for variable 'n'. Reorder the terms: -31 + 12n + -1n2 = 31 + n2 + -31 + -1n2 Reorder the terms: -31 + 12n + -1n2 = 31 + -31 + n2 + -1n2 Combine like terms: 31 + -31 = 0 -31 + 12n + -1n2 = 0 + n2 + -1n2 -31 + 12n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 -31 + 12n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 31 + -12n + n2 = 0 Move the constant term to the right: Add '-31' to each side of the equation. 31 + -12n + -31 + n2 = 0 + -31 Reorder the terms: 31 + -31 + -12n + n2 = 0 + -31 Combine like terms: 31 + -31 = 0 0 + -12n + n2 = 0 + -31 -12n + n2 = 0 + -31 Combine like terms: 0 + -31 = -31 -12n + n2 = -31 The n term is -12n. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12n + 36 + n2 = -31 + 36 Reorder the terms: 36 + -12n + n2 = -31 + 36 Combine like terms: -31 + 36 = 5 36 + -12n + n2 = 5 Factor a perfect square on the left side: (n + -6)(n + -6) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (n + -6) equal to 2.236067978 and -2.236067978.Subproblem 1
n + -6 = 2.236067978 Simplifying n + -6 = 2.236067978 Reorder the terms: -6 + n = 2.236067978 Solving -6 + n = 2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + n = 2.236067978 + 6 n = 2.236067978 + 6 Combine like terms: 2.236067978 + 6 = 8.236067978 n = 8.236067978 Simplifying n = 8.236067978Subproblem 2
n + -6 = -2.236067978 Simplifying n + -6 = -2.236067978 Reorder the terms: -6 + n = -2.236067978 Solving -6 + n = -2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + n = -2.236067978 + 6 n = -2.236067978 + 6 Combine like terms: -2.236067978 + 6 = 3.763932022 n = 3.763932022 Simplifying n = 3.763932022Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.236067978, 3.763932022}
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