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Simplifying (8n + -3) + (4n2 + 3n + -3) = 0 Reorder the terms: (-3 + 8n) + (4n2 + 3n + -3) = 0 Remove parenthesis around (-3 + 8n) -3 + 8n + (4n2 + 3n + -3) = 0 Reorder the terms: -3 + 8n + (-3 + 3n + 4n2) = 0 Remove parenthesis around (-3 + 3n + 4n2) -3 + 8n + -3 + 3n + 4n2 = 0 Reorder the terms: -3 + -3 + 8n + 3n + 4n2 = 0 Combine like terms: -3 + -3 = -6 -6 + 8n + 3n + 4n2 = 0 Combine like terms: 8n + 3n = 11n -6 + 11n + 4n2 = 0 Solving -6 + 11n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.5 + 2.75n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 2.75n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 2.75n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 2.75n + n2 = 0 + 1.5 2.75n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 2.75n + n2 = 1.5 The n term is 2.75n. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75n + 1.890625 + n2 = 1.5 + 1.890625 Reorder the terms: 1.890625 + 2.75n + n2 = 1.5 + 1.890625 Combine like terms: 1.5 + 1.890625 = 3.390625 1.890625 + 2.75n + n2 = 3.390625 Factor a perfect square on the left side: (n + 1.375)(n + 1.375) = 3.390625 Calculate the square root of the right side: 1.841364983 Break this problem into two subproblems by setting (n + 1.375) equal to 1.841364983 and -1.841364983.Subproblem 1
n + 1.375 = 1.841364983 Simplifying n + 1.375 = 1.841364983 Reorder the terms: 1.375 + n = 1.841364983 Solving 1.375 + n = 1.841364983 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + n = 1.841364983 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + n = 1.841364983 + -1.375 n = 1.841364983 + -1.375 Combine like terms: 1.841364983 + -1.375 = 0.466364983 n = 0.466364983 Simplifying n = 0.466364983Subproblem 2
n + 1.375 = -1.841364983 Simplifying n + 1.375 = -1.841364983 Reorder the terms: 1.375 + n = -1.841364983 Solving 1.375 + n = -1.841364983 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + n = -1.841364983 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + n = -1.841364983 + -1.375 n = -1.841364983 + -1.375 Combine like terms: -1.841364983 + -1.375 = -3.216364983 n = -3.216364983 Simplifying n = -3.216364983Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.466364983, -3.216364983}
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