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Simplifying 9n2 + 18n + -61 = 0 Reorder the terms: -61 + 18n + 9n2 = 0 Solving -61 + 18n + 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'. -6.777777778 + 2n + n2 = 0 Move the constant term to the right: Add '6.777777778' to each side of the equation. -6.777777778 + 2n + 6.777777778 + n2 = 0 + 6.777777778 Reorder the terms: -6.777777778 + 6.777777778 + 2n + n2 = 0 + 6.777777778 Combine like terms: -6.777777778 + 6.777777778 = 0.000000000 0.000000000 + 2n + n2 = 0 + 6.777777778 2n + n2 = 0 + 6.777777778 Combine like terms: 0 + 6.777777778 = 6.777777778 2n + n2 = 6.777777778 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 6.777777778 + 1 Reorder the terms: 1 + 2n + n2 = 6.777777778 + 1 Combine like terms: 6.777777778 + 1 = 7.777777778 1 + 2n + n2 = 7.777777778 Factor a perfect square on the left side: (n + 1)(n + 1) = 7.777777778 Calculate the square root of the right side: 2.788866755 Break this problem into two subproblems by setting (n + 1) equal to 2.788866755 and -2.788866755.Subproblem 1
n + 1 = 2.788866755 Simplifying n + 1 = 2.788866755 Reorder the terms: 1 + n = 2.788866755 Solving 1 + n = 2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 2.788866755 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.788866755 + -1 n = 2.788866755 + -1 Combine like terms: 2.788866755 + -1 = 1.788866755 n = 1.788866755 Simplifying n = 1.788866755Subproblem 2
n + 1 = -2.788866755 Simplifying n + 1 = -2.788866755 Reorder the terms: 1 + n = -2.788866755 Solving 1 + n = -2.788866755 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -2.788866755 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.788866755 + -1 n = -2.788866755 + -1 Combine like terms: -2.788866755 + -1 = -3.788866755 n = -3.788866755 Simplifying n = -3.788866755Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.788866755, -3.788866755}
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