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Simplifying 3n + -5 = -8n(6 + 5n) Reorder the terms: -5 + 3n = -8n(6 + 5n) -5 + 3n = (6 * -8n + 5n * -8n) -5 + 3n = (-48n + -40n2) Solving -5 + 3n = -48n + -40n2 Solving for variable 'n'. Combine like terms: 3n + 48n = 51n -5 + 51n + 40n2 = -48n + -40n2 + 48n + 40n2 Reorder the terms: -5 + 51n + 40n2 = -48n + 48n + -40n2 + 40n2 Combine like terms: -48n + 48n = 0 -5 + 51n + 40n2 = 0 + -40n2 + 40n2 -5 + 51n + 40n2 = -40n2 + 40n2 Combine like terms: -40n2 + 40n2 = 0 -5 + 51n + 40n2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.125 + 1.275n + n2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 1.275n + 0.125 + n2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 1.275n + n2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 1.275n + n2 = 0 + 0.125 1.275n + n2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 1.275n + n2 = 0.125 The n term is 1.275n. Take half its coefficient (0.6375). Square it (0.40640625) and add it to both sides. Add '0.40640625' to each side of the equation. 1.275n + 0.40640625 + n2 = 0.125 + 0.40640625 Reorder the terms: 0.40640625 + 1.275n + n2 = 0.125 + 0.40640625 Combine like terms: 0.125 + 0.40640625 = 0.53140625 0.40640625 + 1.275n + n2 = 0.53140625 Factor a perfect square on the left side: (n + 0.6375)(n + 0.6375) = 0.53140625 Calculate the square root of the right side: 0.728976166 Break this problem into two subproblems by setting (n + 0.6375) equal to 0.728976166 and -0.728976166.Subproblem 1
n + 0.6375 = 0.728976166 Simplifying n + 0.6375 = 0.728976166 Reorder the terms: 0.6375 + n = 0.728976166 Solving 0.6375 + n = 0.728976166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6375' to each side of the equation. 0.6375 + -0.6375 + n = 0.728976166 + -0.6375 Combine like terms: 0.6375 + -0.6375 = 0.0000 0.0000 + n = 0.728976166 + -0.6375 n = 0.728976166 + -0.6375 Combine like terms: 0.728976166 + -0.6375 = 0.091476166 n = 0.091476166 Simplifying n = 0.091476166Subproblem 2
n + 0.6375 = -0.728976166 Simplifying n + 0.6375 = -0.728976166 Reorder the terms: 0.6375 + n = -0.728976166 Solving 0.6375 + n = -0.728976166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6375' to each side of the equation. 0.6375 + -0.6375 + n = -0.728976166 + -0.6375 Combine like terms: 0.6375 + -0.6375 = 0.0000 0.0000 + n = -0.728976166 + -0.6375 n = -0.728976166 + -0.6375 Combine like terms: -0.728976166 + -0.6375 = -1.366476166 n = -1.366476166 Simplifying n = -1.366476166Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.091476166, -1.366476166}
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