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Simplifying 9n + 4 + 9n + 4 + 3n2 + -0.75 = 180 Reorder the terms: 4 + 4 + -0.75 + 9n + 9n + 3n2 = 180 Combine like terms: 4 + 4 = 8 8 + -0.75 + 9n + 9n + 3n2 = 180 Combine like terms: 8 + -0.75 = 7.25 7.25 + 9n + 9n + 3n2 = 180 Combine like terms: 9n + 9n = 18n 7.25 + 18n + 3n2 = 180 Solving 7.25 + 18n + 3n2 = 180 Solving for variable 'n'. Reorder the terms: 7.25 + -180 + 18n + 3n2 = 180 + -180 Combine like terms: 7.25 + -180 = -172.75 -172.75 + 18n + 3n2 = 180 + -180 Combine like terms: 180 + -180 = 0 -172.75 + 18n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -57.58333333 + 6n + n2 = 0 Move the constant term to the right: Add '57.58333333' to each side of the equation. -57.58333333 + 6n + 57.58333333 + n2 = 0 + 57.58333333 Reorder the terms: -57.58333333 + 57.58333333 + 6n + n2 = 0 + 57.58333333 Combine like terms: -57.58333333 + 57.58333333 = 0.00000000 0.00000000 + 6n + n2 = 0 + 57.58333333 6n + n2 = 0 + 57.58333333 Combine like terms: 0 + 57.58333333 = 57.58333333 6n + n2 = 57.58333333 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 = 57.58333333 + 9 Reorder the terms: 9 + 6n + n2 = 57.58333333 + 9 Combine like terms: 57.58333333 + 9 = 66.58333333 9 + 6n + n2 = 66.58333333 Factor a perfect square on the left side: (n + 3)(n + 3) = 66.58333333 Calculate the square root of the right side: 8.15986111 Break this problem into two subproblems by setting (n + 3) equal to 8.15986111 and -8.15986111.Subproblem 1
n + 3 = 8.15986111 Simplifying n + 3 = 8.15986111 Reorder the terms: 3 + n = 8.15986111 Solving 3 + n = 8.15986111 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 = 8.15986111 + -3 Combine like terms: 3 + -3 = 0 0 + n = 8.15986111 + -3 n = 8.15986111 + -3 Combine like terms: 8.15986111 + -3 = 5.15986111 n = 5.15986111 Simplifying n = 5.15986111Subproblem 2
n + 3 = -8.15986111 Simplifying n + 3 = -8.15986111 Reorder the terms: 3 + n = -8.15986111 Solving 3 + n = -8.15986111 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 = -8.15986111 + -3 Combine like terms: 3 + -3 = 0 0 + n = -8.15986111 + -3 n = -8.15986111 + -3 Combine like terms: -8.15986111 + -3 = -11.15986111 n = -11.15986111 Simplifying n = -11.15986111Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.15986111, -11.15986111}
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