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Simplifying 12 + -3n + 4(2n + 5) * n = 14 Reorder the terms: 12 + -3n + 4(5 + 2n) * n = 14 Reorder the terms for easier multiplication: 12 + -3n + 4n(5 + 2n) = 14 12 + -3n + (5 * 4n + 2n * 4n) = 14 12 + -3n + (20n + 8n2) = 14 Combine like terms: -3n + 20n = 17n 12 + 17n + 8n2 = 14 Solving 12 + 17n + 8n2 = 14 Solving for variable 'n'. Reorder the terms: 12 + -14 + 17n + 8n2 = 14 + -14 Combine like terms: 12 + -14 = -2 -2 + 17n + 8n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -2 + 17n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.25 + 2.125n + n2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 2.125n + 0.25 + n2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 2.125n + n2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 2.125n + n2 = 0 + 0.25 2.125n + n2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 2.125n + n2 = 0.25 The n term is 2.125n. Take half its coefficient (1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. 2.125n + 1.12890625 + n2 = 0.25 + 1.12890625 Reorder the terms: 1.12890625 + 2.125n + n2 = 0.25 + 1.12890625 Combine like terms: 0.25 + 1.12890625 = 1.37890625 1.12890625 + 2.125n + n2 = 1.37890625 Factor a perfect square on the left side: (n + 1.0625)(n + 1.0625) = 1.37890625 Calculate the square root of the right side: 1.174268389 Break this problem into two subproblems by setting (n + 1.0625) equal to 1.174268389 and -1.174268389.Subproblem 1
n + 1.0625 = 1.174268389 Simplifying n + 1.0625 = 1.174268389 Reorder the terms: 1.0625 + n = 1.174268389 Solving 1.0625 + n = 1.174268389 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + n = 1.174268389 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + n = 1.174268389 + -1.0625 n = 1.174268389 + -1.0625 Combine like terms: 1.174268389 + -1.0625 = 0.111768389 n = 0.111768389 Simplifying n = 0.111768389Subproblem 2
n + 1.0625 = -1.174268389 Simplifying n + 1.0625 = -1.174268389 Reorder the terms: 1.0625 + n = -1.174268389 Solving 1.0625 + n = -1.174268389 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + n = -1.174268389 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + n = -1.174268389 + -1.0625 n = -1.174268389 + -1.0625 Combine like terms: -1.174268389 + -1.0625 = -2.236768389 n = -2.236768389 Simplifying n = -2.236768389Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.111768389, -2.236768389}
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