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Simplifying 9(n + 2) * n + (n + 4) = 7 Reorder the terms: 9(2 + n) * n + (n + 4) = 7 Reorder the terms for easier multiplication: 9n(2 + n) + (n + 4) = 7 (2 * 9n + n * 9n) + (n + 4) = 7 (18n + 9n2) + (n + 4) = 7 Reorder the terms: 18n + 9n2 + (4 + n) = 7 Remove parenthesis around (4 + n) 18n + 9n2 + 4 + n = 7 Reorder the terms: 4 + 18n + n + 9n2 = 7 Combine like terms: 18n + n = 19n 4 + 19n + 9n2 = 7 Solving 4 + 19n + 9n2 = 7 Solving for variable 'n'. Reorder the terms: 4 + -7 + 19n + 9n2 = 7 + -7 Combine like terms: 4 + -7 = -3 -3 + 19n + 9n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -3 + 19n + 9n2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.3333333333 + 2.111111111n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 2.111111111n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 2.111111111n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 2.111111111n + n2 = 0 + 0.3333333333 2.111111111n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 2.111111111n + n2 = 0.3333333333 The n term is 2.111111111n. Take half its coefficient (1.055555556). Square it (1.114197532) and add it to both sides. Add '1.114197532' to each side of the equation. 2.111111111n + 1.114197532 + n2 = 0.3333333333 + 1.114197532 Reorder the terms: 1.114197532 + 2.111111111n + n2 = 0.3333333333 + 1.114197532 Combine like terms: 0.3333333333 + 1.114197532 = 1.4475308653 1.114197532 + 2.111111111n + n2 = 1.4475308653 Factor a perfect square on the left side: (n + 1.055555556)(n + 1.055555556) = 1.4475308653 Calculate the square root of the right side: 1.203133769 Break this problem into two subproblems by setting (n + 1.055555556) equal to 1.203133769 and -1.203133769.Subproblem 1
n + 1.055555556 = 1.203133769 Simplifying n + 1.055555556 = 1.203133769 Reorder the terms: 1.055555556 + n = 1.203133769 Solving 1.055555556 + n = 1.203133769 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + n = 1.203133769 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + n = 1.203133769 + -1.055555556 n = 1.203133769 + -1.055555556 Combine like terms: 1.203133769 + -1.055555556 = 0.147578213 n = 0.147578213 Simplifying n = 0.147578213Subproblem 2
n + 1.055555556 = -1.203133769 Simplifying n + 1.055555556 = -1.203133769 Reorder the terms: 1.055555556 + n = -1.203133769 Solving 1.055555556 + n = -1.203133769 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.055555556' to each side of the equation. 1.055555556 + -1.055555556 + n = -1.203133769 + -1.055555556 Combine like terms: 1.055555556 + -1.055555556 = 0.000000000 0.000000000 + n = -1.203133769 + -1.055555556 n = -1.203133769 + -1.055555556 Combine like terms: -1.203133769 + -1.055555556 = -2.258689325 n = -2.258689325 Simplifying n = -2.258689325Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.147578213, -2.258689325}
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