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Simplifying 4n(6n2 + 3n + -2) = 0 Reorder the terms: 4n(-2 + 3n + 6n2) = 0 (-2 * 4n + 3n * 4n + 6n2 * 4n) = 0 (-8n + 12n2 + 24n3) = 0 Solving -8n + 12n2 + 24n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4n'. 4n(-2 + 3n + 6n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(-2 + 3n + 6n2)' equal to zero and attempt to solve: Simplifying -2 + 3n + 6n2 = 0 Solving -2 + 3n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.3333333333 + 0.5n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.5n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.5n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.5n + n2 = 0 + 0.3333333333 0.5n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.5n + n2 = 0.3333333333 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 0.3333333333 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 0.3333333333 + 0.0625 Combine like terms: 0.3333333333 + 0.0625 = 0.3958333333 0.0625 + 0.5n + n2 = 0.3958333333 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 0.3958333333 Calculate the square root of the right side: 0.62915287 Break this problem into two subproblems by setting (n + 0.25) equal to 0.62915287 and -0.62915287.Subproblem 1
n + 0.25 = 0.62915287 Simplifying n + 0.25 = 0.62915287 Reorder the terms: 0.25 + n = 0.62915287 Solving 0.25 + n = 0.62915287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 0.62915287 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 0.62915287 + -0.25 n = 0.62915287 + -0.25 Combine like terms: 0.62915287 + -0.25 = 0.37915287 n = 0.37915287 Simplifying n = 0.37915287Subproblem 2
n + 0.25 = -0.62915287 Simplifying n + 0.25 = -0.62915287 Reorder the terms: 0.25 + n = -0.62915287 Solving 0.25 + n = -0.62915287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -0.62915287 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -0.62915287 + -0.25 n = -0.62915287 + -0.25 Combine like terms: -0.62915287 + -0.25 = -0.87915287 n = -0.87915287 Simplifying n = -0.87915287Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.37915287, -0.87915287}Solution
n = {0, 0.37915287, -0.87915287}
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