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Simplifying n(2n + 3) + 4(2n + -3) = 0 Reorder the terms: n(3 + 2n) + 4(2n + -3) = 0 (3 * n + 2n * n) + 4(2n + -3) = 0 (3n + 2n2) + 4(2n + -3) = 0 Reorder the terms: 3n + 2n2 + 4(-3 + 2n) = 0 3n + 2n2 + (-3 * 4 + 2n * 4) = 0 3n + 2n2 + (-12 + 8n) = 0 Reorder the terms: -12 + 3n + 8n + 2n2 = 0 Combine like terms: 3n + 8n = 11n -12 + 11n + 2n2 = 0 Solving -12 + 11n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + 5.5n + n2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 5.5n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + 5.5n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 5.5n + n2 = 0 + 6 5.5n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 5.5n + n2 = 6 The n term is 5.5n. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5n + 7.5625 + n2 = 6 + 7.5625 Reorder the terms: 7.5625 + 5.5n + n2 = 6 + 7.5625 Combine like terms: 6 + 7.5625 = 13.5625 7.5625 + 5.5n + n2 = 13.5625 Factor a perfect square on the left side: (n + 2.75)(n + 2.75) = 13.5625 Calculate the square root of the right side: 3.682729966 Break this problem into two subproblems by setting (n + 2.75) equal to 3.682729966 and -3.682729966.Subproblem 1
n + 2.75 = 3.682729966 Simplifying n + 2.75 = 3.682729966 Reorder the terms: 2.75 + n = 3.682729966 Solving 2.75 + n = 3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + n = 3.682729966 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = 3.682729966 + -2.75 n = 3.682729966 + -2.75 Combine like terms: 3.682729966 + -2.75 = 0.932729966 n = 0.932729966 Simplifying n = 0.932729966Subproblem 2
n + 2.75 = -3.682729966 Simplifying n + 2.75 = -3.682729966 Reorder the terms: 2.75 + n = -3.682729966 Solving 2.75 + n = -3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + n = -3.682729966 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = -3.682729966 + -2.75 n = -3.682729966 + -2.75 Combine like terms: -3.682729966 + -2.75 = -6.432729966 n = -6.432729966 Simplifying n = -6.432729966Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.932729966, -6.432729966}
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