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Simplifying 0 = 5n2 + 38n + 16 Reorder the terms: 0 = 16 + 38n + 5n2 Solving 0 = 16 + 38n + 5n2 Solving for variable 'n'. Combine like terms: 0 + -16 = -16 -16 + -38n + -5n2 = 16 + 38n + 5n2 + -16 + -38n + -5n2 Reorder the terms: -16 + -38n + -5n2 = 16 + -16 + 38n + -38n + 5n2 + -5n2 Combine like terms: 16 + -16 = 0 -16 + -38n + -5n2 = 0 + 38n + -38n + 5n2 + -5n2 -16 + -38n + -5n2 = 38n + -38n + 5n2 + -5n2 Combine like terms: 38n + -38n = 0 -16 + -38n + -5n2 = 0 + 5n2 + -5n2 -16 + -38n + -5n2 = 5n2 + -5n2 Combine like terms: 5n2 + -5n2 = 0 -16 + -38n + -5n2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(16 + 38n + 5n2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(16 + 38n + 5n2)' equal to zero and attempt to solve: Simplifying 16 + 38n + 5n2 = 0 Solving 16 + 38n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 3.2 + 7.6n + n2 = 0 Move the constant term to the right: Add '-3.2' to each side of the equation. 3.2 + 7.6n + -3.2 + n2 = 0 + -3.2 Reorder the terms: 3.2 + -3.2 + 7.6n + n2 = 0 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + 7.6n + n2 = 0 + -3.2 7.6n + n2 = 0 + -3.2 Combine like terms: 0 + -3.2 = -3.2 7.6n + n2 = -3.2 The n term is 7.6n. Take half its coefficient (3.8). Square it (14.44) and add it to both sides. Add '14.44' to each side of the equation. 7.6n + 14.44 + n2 = -3.2 + 14.44 Reorder the terms: 14.44 + 7.6n + n2 = -3.2 + 14.44 Combine like terms: -3.2 + 14.44 = 11.24 14.44 + 7.6n + n2 = 11.24 Factor a perfect square on the left side: (n + 3.8)(n + 3.8) = 11.24 Calculate the square root of the right side: 3.352610923 Break this problem into two subproblems by setting (n + 3.8) equal to 3.352610923 and -3.352610923.Subproblem 1
n + 3.8 = 3.352610923 Simplifying n + 3.8 = 3.352610923 Reorder the terms: 3.8 + n = 3.352610923 Solving 3.8 + n = 3.352610923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.8' to each side of the equation. 3.8 + -3.8 + n = 3.352610923 + -3.8 Combine like terms: 3.8 + -3.8 = 0.0 0.0 + n = 3.352610923 + -3.8 n = 3.352610923 + -3.8 Combine like terms: 3.352610923 + -3.8 = -0.447389077 n = -0.447389077 Simplifying n = -0.447389077Subproblem 2
n + 3.8 = -3.352610923 Simplifying n + 3.8 = -3.352610923 Reorder the terms: 3.8 + n = -3.352610923 Solving 3.8 + n = -3.352610923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.8' to each side of the equation. 3.8 + -3.8 + n = -3.352610923 + -3.8 Combine like terms: 3.8 + -3.8 = 0.0 0.0 + n = -3.352610923 + -3.8 n = -3.352610923 + -3.8 Combine like terms: -3.352610923 + -3.8 = -7.152610923 n = -7.152610923 Simplifying n = -7.152610923Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.447389077, -7.152610923}Solution
n = {-0.447389077, -7.152610923}
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