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Simplifying 24n3 + 15n2 + -15n = 0 Reorder the terms: -15n + 15n2 + 24n3 = 0 Solving -15n + 15n2 + 24n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3n'. 3n(-5 + 5n + 8n2) = 0 Ignore the factor 3.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 '(-5 + 5n + 8n2)' equal to zero and attempt to solve: Simplifying -5 + 5n + 8n2 = 0 Solving -5 + 5n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.625 + 0.625n + n2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 0.625n + 0.625 + n2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 0.625n + n2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 0.625n + n2 = 0 + 0.625 0.625n + n2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 0.625n + n2 = 0.625 The n term is 0.625n. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625n + 0.09765625 + n2 = 0.625 + 0.09765625 Reorder the terms: 0.09765625 + 0.625n + n2 = 0.625 + 0.09765625 Combine like terms: 0.625 + 0.09765625 = 0.72265625 0.09765625 + 0.625n + n2 = 0.72265625 Factor a perfect square on the left side: (n + 0.3125)(n + 0.3125) = 0.72265625 Calculate the square root of the right side: 0.850091907 Break this problem into two subproblems by setting (n + 0.3125) equal to 0.850091907 and -0.850091907.Subproblem 1
n + 0.3125 = 0.850091907 Simplifying n + 0.3125 = 0.850091907 Reorder the terms: 0.3125 + n = 0.850091907 Solving 0.3125 + n = 0.850091907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + n = 0.850091907 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + n = 0.850091907 + -0.3125 n = 0.850091907 + -0.3125 Combine like terms: 0.850091907 + -0.3125 = 0.537591907 n = 0.537591907 Simplifying n = 0.537591907Subproblem 2
n + 0.3125 = -0.850091907 Simplifying n + 0.3125 = -0.850091907 Reorder the terms: 0.3125 + n = -0.850091907 Solving 0.3125 + n = -0.850091907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + n = -0.850091907 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + n = -0.850091907 + -0.3125 n = -0.850091907 + -0.3125 Combine like terms: -0.850091907 + -0.3125 = -1.162591907 n = -1.162591907 Simplifying n = -1.162591907Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.537591907, -1.162591907}Solution
n = {0, 0.537591907, -1.162591907}
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