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Simplifying 16n2 + -114n = 14 Reorder the terms: -114n + 16n2 = 14 Solving -114n + 16n2 = 14 Solving for variable 'n'. Reorder the terms: -14 + -114n + 16n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + -114n + 16n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -57n + 8n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -57n + 8n2)' equal to zero and attempt to solve: Simplifying -7 + -57n + 8n2 = 0 Solving -7 + -57n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + -7.125n + n2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -7.125n + 0.875 + n2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -7.125n + n2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -7.125n + n2 = 0 + 0.875 -7.125n + n2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -7.125n + n2 = 0.875 The n term is -7.125n. Take half its coefficient (-3.5625). Square it (12.69140625) and add it to both sides. Add '12.69140625' to each side of the equation. -7.125n + 12.69140625 + n2 = 0.875 + 12.69140625 Reorder the terms: 12.69140625 + -7.125n + n2 = 0.875 + 12.69140625 Combine like terms: 0.875 + 12.69140625 = 13.56640625 12.69140625 + -7.125n + n2 = 13.56640625 Factor a perfect square on the left side: (n + -3.5625)(n + -3.5625) = 13.56640625 Calculate the square root of the right side: 3.683260275 Break this problem into two subproblems by setting (n + -3.5625) equal to 3.683260275 and -3.683260275.Subproblem 1
n + -3.5625 = 3.683260275 Simplifying n + -3.5625 = 3.683260275 Reorder the terms: -3.5625 + n = 3.683260275 Solving -3.5625 + n = 3.683260275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5625' to each side of the equation. -3.5625 + 3.5625 + n = 3.683260275 + 3.5625 Combine like terms: -3.5625 + 3.5625 = 0.0000 0.0000 + n = 3.683260275 + 3.5625 n = 3.683260275 + 3.5625 Combine like terms: 3.683260275 + 3.5625 = 7.245760275 n = 7.245760275 Simplifying n = 7.245760275Subproblem 2
n + -3.5625 = -3.683260275 Simplifying n + -3.5625 = -3.683260275 Reorder the terms: -3.5625 + n = -3.683260275 Solving -3.5625 + n = -3.683260275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.5625' to each side of the equation. -3.5625 + 3.5625 + n = -3.683260275 + 3.5625 Combine like terms: -3.5625 + 3.5625 = 0.0000 0.0000 + n = -3.683260275 + 3.5625 n = -3.683260275 + 3.5625 Combine like terms: -3.683260275 + 3.5625 = -0.120760275 n = -0.120760275 Simplifying n = -0.120760275Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.245760275, -0.120760275}Solution
n = {7.245760275, -0.120760275}
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