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Simplifying 3n + 5 = 2n2 + -16 Reorder the terms: 5 + 3n = 2n2 + -16 Reorder the terms: 5 + 3n = -16 + 2n2 Solving 5 + 3n = -16 + 2n2 Solving for variable 'n'. Reorder the terms: 5 + 16 + 3n + -2n2 = -16 + 2n2 + 16 + -2n2 Combine like terms: 5 + 16 = 21 21 + 3n + -2n2 = -16 + 2n2 + 16 + -2n2 Reorder the terms: 21 + 3n + -2n2 = -16 + 16 + 2n2 + -2n2 Combine like terms: -16 + 16 = 0 21 + 3n + -2n2 = 0 + 2n2 + -2n2 21 + 3n + -2n2 = 2n2 + -2n2 Combine like terms: 2n2 + -2n2 = 0 21 + 3n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -10.5 + -1.5n + n2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -1.5n + 10.5 + n2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -1.5n + n2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -1.5n + n2 = 0 + 10.5 -1.5n + n2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -1.5n + n2 = 10.5 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 10.5 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 10.5 + 0.5625 Combine like terms: 10.5 + 0.5625 = 11.0625 0.5625 + -1.5n + n2 = 11.0625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 11.0625 Calculate the square root of the right side: 3.326033674 Break this problem into two subproblems by setting (n + -0.75) equal to 3.326033674 and -3.326033674.Subproblem 1
n + -0.75 = 3.326033674 Simplifying n + -0.75 = 3.326033674 Reorder the terms: -0.75 + n = 3.326033674 Solving -0.75 + n = 3.326033674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 3.326033674 + 0.75 n = 3.326033674 + 0.75 Combine like terms: 3.326033674 + 0.75 = 4.076033674 n = 4.076033674 Simplifying n = 4.076033674Subproblem 2
n + -0.75 = -3.326033674 Simplifying n + -0.75 = -3.326033674 Reorder the terms: -0.75 + n = -3.326033674 Solving -0.75 + n = -3.326033674 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -3.326033674 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -3.326033674 + 0.75 n = -3.326033674 + 0.75 Combine like terms: -3.326033674 + 0.75 = -2.576033674 n = -2.576033674 Simplifying n = -2.576033674Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.076033674, -2.576033674}
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