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Simplifying 6n2 + 30n + -9 = 0 Reorder the terms: -9 + 30n + 6n2 = 0 Solving -9 + 30n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-3 + 10n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-3 + 10n + 2n2)' equal to zero and attempt to solve: Simplifying -3 + 10n + 2n2 = 0 Solving -3 + 10n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 5n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 5n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 5n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 5n + n2 = 0 + 1.5 5n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 5n + n2 = 1.5 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 1.5 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 1.5 + 6.25 Combine like terms: 1.5 + 6.25 = 7.75 6.25 + 5n + n2 = 7.75 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (n + 2.5) equal to 2.783882181 and -2.783882181.Subproblem 1
n + 2.5 = 2.783882181 Simplifying n + 2.5 = 2.783882181 Reorder the terms: 2.5 + n = 2.783882181 Solving 2.5 + n = 2.783882181 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 2.783882181 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 2.783882181 + -2.5 n = 2.783882181 + -2.5 Combine like terms: 2.783882181 + -2.5 = 0.283882181 n = 0.283882181 Simplifying n = 0.283882181Subproblem 2
n + 2.5 = -2.783882181 Simplifying n + 2.5 = -2.783882181 Reorder the terms: 2.5 + n = -2.783882181 Solving 2.5 + n = -2.783882181 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -2.783882181 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -2.783882181 + -2.5 n = -2.783882181 + -2.5 Combine like terms: -2.783882181 + -2.5 = -5.283882181 n = -5.283882181 Simplifying n = -5.283882181Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.283882181, -5.283882181}Solution
n = {0.283882181, -5.283882181}
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