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Simplifying 6n2 + -18n + -18 = 15 Reorder the terms: -18 + -18n + 6n2 = 15 Solving -18 + -18n + 6n2 = 15 Solving for variable 'n'. Reorder the terms: -18 + -15 + -18n + 6n2 = 15 + -15 Combine like terms: -18 + -15 = -33 -33 + -18n + 6n2 = 15 + -15 Combine like terms: 15 + -15 = 0 -33 + -18n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-11 + -6n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-11 + -6n + 2n2)' equal to zero and attempt to solve: Simplifying -11 + -6n + 2n2 = 0 Solving -11 + -6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5.5 + -3n + n2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + -3n + 5.5 + n2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + -3n + n2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + -3n + n2 = 0 + 5.5 -3n + n2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 -3n + n2 = 5.5 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 5.5 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 5.5 + 2.25 Combine like terms: 5.5 + 2.25 = 7.75 2.25 + -3n + n2 = 7.75 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (n + -1.5) equal to 2.783882181 and -2.783882181.Subproblem 1
n + -1.5 = 2.783882181 Simplifying n + -1.5 = 2.783882181 Reorder the terms: -1.5 + n = 2.783882181 Solving -1.5 + n = 2.783882181 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 2.783882181 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.783882181 + 1.5 n = 2.783882181 + 1.5 Combine like terms: 2.783882181 + 1.5 = 4.283882181 n = 4.283882181 Simplifying n = 4.283882181Subproblem 2
n + -1.5 = -2.783882181 Simplifying n + -1.5 = -2.783882181 Reorder the terms: -1.5 + n = -2.783882181 Solving -1.5 + n = -2.783882181 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -2.783882181 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.783882181 + 1.5 n = -2.783882181 + 1.5 Combine like terms: -2.783882181 + 1.5 = -1.283882181 n = -1.283882181 Simplifying n = -1.283882181Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.283882181, -1.283882181}Solution
n = {4.283882181, -1.283882181}
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