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Simplifying 28n2 = -96 + -185n Solving 28n2 = -96 + -185n Solving for variable 'n'. Reorder the terms: 96 + 185n + 28n2 = -96 + -185n + 96 + 185n Reorder the terms: 96 + 185n + 28n2 = -96 + 96 + -185n + 185n Combine like terms: -96 + 96 = 0 96 + 185n + 28n2 = 0 + -185n + 185n 96 + 185n + 28n2 = -185n + 185n Combine like terms: -185n + 185n = 0 96 + 185n + 28n2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. 3.428571429 + 6.607142857n + n2 = 0 Move the constant term to the right: Add '-3.428571429' to each side of the equation. 3.428571429 + 6.607142857n + -3.428571429 + n2 = 0 + -3.428571429 Reorder the terms: 3.428571429 + -3.428571429 + 6.607142857n + n2 = 0 + -3.428571429 Combine like terms: 3.428571429 + -3.428571429 = 0.000000000 0.000000000 + 6.607142857n + n2 = 0 + -3.428571429 6.607142857n + n2 = 0 + -3.428571429 Combine like terms: 0 + -3.428571429 = -3.428571429 6.607142857n + n2 = -3.428571429 The n term is 6.607142857n. Take half its coefficient (3.303571429). Square it (10.91358419) and add it to both sides. Add '10.91358419' to each side of the equation. 6.607142857n + 10.91358419 + n2 = -3.428571429 + 10.91358419 Reorder the terms: 10.91358419 + 6.607142857n + n2 = -3.428571429 + 10.91358419 Combine like terms: -3.428571429 + 10.91358419 = 7.485012761 10.91358419 + 6.607142857n + n2 = 7.485012761 Factor a perfect square on the left side: (n + 3.303571429)(n + 3.303571429) = 7.485012761 Calculate the square root of the right side: 2.735875136 Break this problem into two subproblems by setting (n + 3.303571429) equal to 2.735875136 and -2.735875136.Subproblem 1
n + 3.303571429 = 2.735875136 Simplifying n + 3.303571429 = 2.735875136 Reorder the terms: 3.303571429 + n = 2.735875136 Solving 3.303571429 + n = 2.735875136 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.303571429' to each side of the equation. 3.303571429 + -3.303571429 + n = 2.735875136 + -3.303571429 Combine like terms: 3.303571429 + -3.303571429 = 0.000000000 0.000000000 + n = 2.735875136 + -3.303571429 n = 2.735875136 + -3.303571429 Combine like terms: 2.735875136 + -3.303571429 = -0.567696293 n = -0.567696293 Simplifying n = -0.567696293Subproblem 2
n + 3.303571429 = -2.735875136 Simplifying n + 3.303571429 = -2.735875136 Reorder the terms: 3.303571429 + n = -2.735875136 Solving 3.303571429 + n = -2.735875136 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.303571429' to each side of the equation. 3.303571429 + -3.303571429 + n = -2.735875136 + -3.303571429 Combine like terms: 3.303571429 + -3.303571429 = 0.000000000 0.000000000 + n = -2.735875136 + -3.303571429 n = -2.735875136 + -3.303571429 Combine like terms: -2.735875136 + -3.303571429 = -6.039446565 n = -6.039446565 Simplifying n = -6.039446565Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.567696293, -6.039446565}
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