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Simplifying (9n2) + 18n + -43 = 0 Reorder the terms: -43 + 18n + (9n2) = 0 Solving -43 + 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'. -4.777777778 + 2n + n2 = 0 Move the constant term to the right: Add '4.777777778' to each side of the equation. -4.777777778 + 2n + 4.777777778 + n2 = 0 + 4.777777778 Reorder the terms: -4.777777778 + 4.777777778 + 2n + n2 = 0 + 4.777777778 Combine like terms: -4.777777778 + 4.777777778 = 0.000000000 0.000000000 + 2n + n2 = 0 + 4.777777778 2n + n2 = 0 + 4.777777778 Combine like terms: 0 + 4.777777778 = 4.777777778 2n + n2 = 4.777777778 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 = 4.777777778 + 1 Reorder the terms: 1 + 2n + n2 = 4.777777778 + 1 Combine like terms: 4.777777778 + 1 = 5.777777778 1 + 2n + n2 = 5.777777778 Factor a perfect square on the left side: ((n) + 1)((n) + 1) = 5.777777778 Calculate the square root of the right side: 2.40370085 Break this problem into two subproblems by setting ((n) + 1) equal to 2.40370085 and -2.40370085.Subproblem 1
(n) + 1 = 2.40370085 Simplifying (n) + 1 = 2.40370085 n + 1 = 2.40370085 Reorder the terms: 1 + n = 2.40370085 Solving 1 + n = 2.40370085 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 = 2.40370085 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.40370085 + -1 n = 2.40370085 + -1 Combine like terms: 2.40370085 + -1 = 1.40370085 n = 1.40370085 Simplifying n = 1.40370085Subproblem 2
(n) + 1 = -2.40370085 Simplifying (n) + 1 = -2.40370085 n + 1 = -2.40370085 Reorder the terms: 1 + n = -2.40370085 Solving 1 + n = -2.40370085 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 = -2.40370085 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.40370085 + -1 n = -2.40370085 + -1 Combine like terms: -2.40370085 + -1 = -3.40370085 n = -3.40370085 Simplifying n = -3.40370085Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.40370085, -3.40370085}
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