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Simplifying 6n3 + 21n2 + -15n = 0 Reorder the terms: -15n + 21n2 + 6n3 = 0 Solving -15n + 21n2 + 6n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3n'. 3n(-5 + 7n + 2n2) = 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 + 7n + 2n2)' equal to zero and attempt to solve: Simplifying -5 + 7n + 2n2 = 0 Solving -5 + 7n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 3.5n + n2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 3.5n + 2.5 + n2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 3.5n + n2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 3.5n + n2 = 0 + 2.5 3.5n + n2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 3.5n + n2 = 2.5 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = 2.5 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = 2.5 + 3.0625 Combine like terms: 2.5 + 3.0625 = 5.5625 3.0625 + 3.5n + n2 = 5.5625 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (n + 1.75) equal to 2.358495283 and -2.358495283.Subproblem 1
n + 1.75 = 2.358495283 Simplifying n + 1.75 = 2.358495283 Reorder the terms: 1.75 + n = 2.358495283 Solving 1.75 + n = 2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 2.358495283 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 2.358495283 + -1.75 n = 2.358495283 + -1.75 Combine like terms: 2.358495283 + -1.75 = 0.608495283 n = 0.608495283 Simplifying n = 0.608495283Subproblem 2
n + 1.75 = -2.358495283 Simplifying n + 1.75 = -2.358495283 Reorder the terms: 1.75 + n = -2.358495283 Solving 1.75 + n = -2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -2.358495283 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -2.358495283 + -1.75 n = -2.358495283 + -1.75 Combine like terms: -2.358495283 + -1.75 = -4.108495283 n = -4.108495283 Simplifying n = -4.108495283Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.608495283, -4.108495283}Solution
n = {0, 0.608495283, -4.108495283}
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