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Simplifying 9n2 + 27n + -20 = 0 Reorder the terms: -20 + 27n + 9n2 = 0 Solving -20 + 27n + 9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.222222222 + 3n + n2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + 3n + 2.222222222 + n2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + 3n + n2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + 3n + n2 = 0 + 2.222222222 3n + n2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 3n + n2 = 2.222222222 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 2.222222222 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 2.222222222 + 2.25 Combine like terms: 2.222222222 + 2.25 = 4.472222222 2.25 + 3n + n2 = 4.472222222 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 4.472222222 Calculate the square root of the right side: 2.114762923 Break this problem into two subproblems by setting (n + 1.5) equal to 2.114762923 and -2.114762923.Subproblem 1
n + 1.5 = 2.114762923 Simplifying n + 1.5 = 2.114762923 Reorder the terms: 1.5 + n = 2.114762923 Solving 1.5 + n = 2.114762923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 2.114762923 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.114762923 + -1.5 n = 2.114762923 + -1.5 Combine like terms: 2.114762923 + -1.5 = 0.614762923 n = 0.614762923 Simplifying n = 0.614762923Subproblem 2
n + 1.5 = -2.114762923 Simplifying n + 1.5 = -2.114762923 Reorder the terms: 1.5 + n = -2.114762923 Solving 1.5 + n = -2.114762923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -2.114762923 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.114762923 + -1.5 n = -2.114762923 + -1.5 Combine like terms: -2.114762923 + -1.5 = -3.614762923 n = -3.614762923 Simplifying n = -3.614762923Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.614762923, -3.614762923}
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