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Simplifying 3n(n2 + 11n + 5) = 0 Reorder the terms: 3n(5 + 11n + n2) = 0 (5 * 3n + 11n * 3n + n2 * 3n) = 0 (15n + 33n2 + 3n3) = 0 Solving 15n + 33n2 + 3n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3n'. 3n(5 + 11n + n2) = 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 + 11n + n2)' equal to zero and attempt to solve: Simplifying 5 + 11n + n2 = 0 Solving 5 + 11n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 11n + -5 + n2 = 0 + -5 Reorder the terms: 5 + -5 + 11n + n2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 11n + n2 = 0 + -5 11n + n2 = 0 + -5 Combine like terms: 0 + -5 = -5 11n + n2 = -5 The n term is 11n. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11n + 30.25 + n2 = -5 + 30.25 Reorder the terms: 30.25 + 11n + n2 = -5 + 30.25 Combine like terms: -5 + 30.25 = 25.25 30.25 + 11n + n2 = 25.25 Factor a perfect square on the left side: (n + 5.5)(n + 5.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (n + 5.5) equal to 5.024937811 and -5.024937811.Subproblem 1
n + 5.5 = 5.024937811 Simplifying n + 5.5 = 5.024937811 Reorder the terms: 5.5 + n = 5.024937811 Solving 5.5 + n = 5.024937811 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = 5.024937811 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = 5.024937811 + -5.5 n = 5.024937811 + -5.5 Combine like terms: 5.024937811 + -5.5 = -0.475062189 n = -0.475062189 Simplifying n = -0.475062189Subproblem 2
n + 5.5 = -5.024937811 Simplifying n + 5.5 = -5.024937811 Reorder the terms: 5.5 + n = -5.024937811 Solving 5.5 + n = -5.024937811 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = -5.024937811 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = -5.024937811 + -5.5 n = -5.024937811 + -5.5 Combine like terms: -5.024937811 + -5.5 = -10.524937811 n = -10.524937811 Simplifying n = -10.524937811Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.475062189, -10.524937811}Solution
n = {0, -0.475062189, -10.524937811}
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