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Simplifying 28n + 4n2 + -4000 = 0 Reorder the terms: -4000 + 28n + 4n2 = 0 Solving -4000 + 28n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(-1000 + 7n + n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1000 + 7n + n2)' equal to zero and attempt to solve: Simplifying -1000 + 7n + n2 = 0 Solving -1000 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 7n + 1000 + n2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 7n + n2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 7n + n2 = 0 + 1000 7n + n2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 7n + n2 = 1000 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 1000 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 1000 + 12.25 Combine like terms: 1000 + 12.25 = 1012.25 12.25 + 7n + n2 = 1012.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 1012.25 Calculate the square root of the right side: 31.81587654 Break this problem into two subproblems by setting (n + 3.5) equal to 31.81587654 and -31.81587654.Subproblem 1
n + 3.5 = 31.81587654 Simplifying n + 3.5 = 31.81587654 Reorder the terms: 3.5 + n = 31.81587654 Solving 3.5 + n = 31.81587654 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 31.81587654 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 31.81587654 + -3.5 n = 31.81587654 + -3.5 Combine like terms: 31.81587654 + -3.5 = 28.31587654 n = 28.31587654 Simplifying n = 28.31587654Subproblem 2
n + 3.5 = -31.81587654 Simplifying n + 3.5 = -31.81587654 Reorder the terms: 3.5 + n = -31.81587654 Solving 3.5 + n = -31.81587654 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -31.81587654 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -31.81587654 + -3.5 n = -31.81587654 + -3.5 Combine like terms: -31.81587654 + -3.5 = -35.31587654 n = -35.31587654 Simplifying n = -35.31587654Solution
The solution to the problem is based on the solutions from the subproblems. n = {28.31587654, -35.31587654}Solution
n = {28.31587654, -35.31587654}
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