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Simplifying 4n2 + 6n + -16 = 0 Reorder the terms: -16 + 6n + 4n2 = 0 Solving -16 + 6n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-8 + 3n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-8 + 3n + 2n2)' equal to zero and attempt to solve: Simplifying -8 + 3n + 2n2 = 0 Solving -8 + 3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4 + 1.5n + n2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 1.5n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + 1.5n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 1.5n + n2 = 0 + 4 1.5n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 1.5n + n2 = 4 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 = 4 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 4 + 0.5625 Combine like terms: 4 + 0.5625 = 4.5625 0.5625 + 1.5n + n2 = 4.5625 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 4.5625 Calculate the square root of the right side: 2.136000936 Break this problem into two subproblems by setting (n + 0.75) equal to 2.136000936 and -2.136000936.Subproblem 1
n + 0.75 = 2.136000936 Simplifying n + 0.75 = 2.136000936 Reorder the terms: 0.75 + n = 2.136000936 Solving 0.75 + n = 2.136000936 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 = 2.136000936 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 2.136000936 + -0.75 n = 2.136000936 + -0.75 Combine like terms: 2.136000936 + -0.75 = 1.386000936 n = 1.386000936 Simplifying n = 1.386000936Subproblem 2
n + 0.75 = -2.136000936 Simplifying n + 0.75 = -2.136000936 Reorder the terms: 0.75 + n = -2.136000936 Solving 0.75 + n = -2.136000936 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 = -2.136000936 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -2.136000936 + -0.75 n = -2.136000936 + -0.75 Combine like terms: -2.136000936 + -0.75 = -2.886000936 n = -2.886000936 Simplifying n = -2.886000936Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.386000936, -2.886000936}Solution
n = {1.386000936, -2.886000936}
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