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Simplifying 0 = 8n2 + -28n + -30 Reorder the terms: 0 = -30 + -28n + 8n2 Solving 0 = -30 + -28n + 8n2 Solving for variable 'n'. Combine like terms: 0 + 30 = 30 30 + 28n + -8n2 = -30 + -28n + 8n2 + 30 + 28n + -8n2 Reorder the terms: 30 + 28n + -8n2 = -30 + 30 + -28n + 28n + 8n2 + -8n2 Combine like terms: -30 + 30 = 0 30 + 28n + -8n2 = 0 + -28n + 28n + 8n2 + -8n2 30 + 28n + -8n2 = -28n + 28n + 8n2 + -8n2 Combine like terms: -28n + 28n = 0 30 + 28n + -8n2 = 0 + 8n2 + -8n2 30 + 28n + -8n2 = 8n2 + -8n2 Combine like terms: 8n2 + -8n2 = 0 30 + 28n + -8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(15 + 14n + -4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(15 + 14n + -4n2)' equal to zero and attempt to solve: Simplifying 15 + 14n + -4n2 = 0 Solving 15 + 14n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -3.75 + -3.5n + n2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + -3.5n + 3.75 + n2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + -3.5n + n2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + -3.5n + n2 = 0 + 3.75 -3.5n + n2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 -3.5n + n2 = 3.75 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 = 3.75 + 3.0625 Reorder the terms: 3.0625 + -3.5n + n2 = 3.75 + 3.0625 Combine like terms: 3.75 + 3.0625 = 6.8125 3.0625 + -3.5n + n2 = 6.8125 Factor a perfect square on the left side: (n + -1.75)(n + -1.75) = 6.8125 Calculate the square root of the right side: 2.610076627 Break this problem into two subproblems by setting (n + -1.75) equal to 2.610076627 and -2.610076627.Subproblem 1
n + -1.75 = 2.610076627 Simplifying n + -1.75 = 2.610076627 Reorder the terms: -1.75 + n = 2.610076627 Solving -1.75 + n = 2.610076627 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 = 2.610076627 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 2.610076627 + 1.75 n = 2.610076627 + 1.75 Combine like terms: 2.610076627 + 1.75 = 4.360076627 n = 4.360076627 Simplifying n = 4.360076627Subproblem 2
n + -1.75 = -2.610076627 Simplifying n + -1.75 = -2.610076627 Reorder the terms: -1.75 + n = -2.610076627 Solving -1.75 + n = -2.610076627 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 = -2.610076627 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -2.610076627 + 1.75 n = -2.610076627 + 1.75 Combine like terms: -2.610076627 + 1.75 = -0.860076627 n = -0.860076627 Simplifying n = -0.860076627Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.360076627, -0.860076627}Solution
n = {4.360076627, -0.860076627}
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