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Simplifying -3(3n + 8) = -8(n + 3) * n Reorder the terms: -3(8 + 3n) = -8(n + 3) * n (8 * -3 + 3n * -3) = -8(n + 3) * n (-24 + -9n) = -8(n + 3) * n Reorder the terms: -24 + -9n = -8(3 + n) * n Reorder the terms for easier multiplication: -24 + -9n = -8n(3 + n) -24 + -9n = (3 * -8n + n * -8n) -24 + -9n = (-24n + -8n2) Solving -24 + -9n = -24n + -8n2 Solving for variable 'n'. Combine like terms: -9n + 24n = 15n -24 + 15n + 8n2 = -24n + -8n2 + 24n + 8n2 Reorder the terms: -24 + 15n + 8n2 = -24n + 24n + -8n2 + 8n2 Combine like terms: -24n + 24n = 0 -24 + 15n + 8n2 = 0 + -8n2 + 8n2 -24 + 15n + 8n2 = -8n2 + 8n2 Combine like terms: -8n2 + 8n2 = 0 -24 + 15n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3 + 1.875n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.875n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + 1.875n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.875n + n2 = 0 + 3 1.875n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.875n + n2 = 3 The n term is 1.875n. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875n + 0.87890625 + n2 = 3 + 0.87890625 Reorder the terms: 0.87890625 + 1.875n + n2 = 3 + 0.87890625 Combine like terms: 3 + 0.87890625 = 3.87890625 0.87890625 + 1.875n + n2 = 3.87890625 Factor a perfect square on the left side: (n + 0.9375)(n + 0.9375) = 3.87890625 Calculate the square root of the right side: 1.969493907 Break this problem into two subproblems by setting (n + 0.9375) equal to 1.969493907 and -1.969493907.Subproblem 1
n + 0.9375 = 1.969493907 Simplifying n + 0.9375 = 1.969493907 Reorder the terms: 0.9375 + n = 1.969493907 Solving 0.9375 + n = 1.969493907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + n = 1.969493907 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + n = 1.969493907 + -0.9375 n = 1.969493907 + -0.9375 Combine like terms: 1.969493907 + -0.9375 = 1.031993907 n = 1.031993907 Simplifying n = 1.031993907Subproblem 2
n + 0.9375 = -1.969493907 Simplifying n + 0.9375 = -1.969493907 Reorder the terms: 0.9375 + n = -1.969493907 Solving 0.9375 + n = -1.969493907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + n = -1.969493907 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + n = -1.969493907 + -0.9375 n = -1.969493907 + -0.9375 Combine like terms: -1.969493907 + -0.9375 = -2.906993907 n = -2.906993907 Simplifying n = -2.906993907Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.031993907, -2.906993907}
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