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Simplifying 4n2 + 28n + -140 = 0 Reorder the terms: -140 + 28n + 4n2 = 0 Solving -140 + 28n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(-35 + 7n + n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-35 + 7n + n2)' equal to zero and attempt to solve: Simplifying -35 + 7n + n2 = 0 Solving -35 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '35' to each side of the equation. -35 + 7n + 35 + n2 = 0 + 35 Reorder the terms: -35 + 35 + 7n + n2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 7n + n2 = 0 + 35 7n + n2 = 0 + 35 Combine like terms: 0 + 35 = 35 7n + n2 = 35 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 = 35 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 35 + 12.25 Combine like terms: 35 + 12.25 = 47.25 12.25 + 7n + n2 = 47.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 47.25 Calculate the square root of the right side: 6.873863542 Break this problem into two subproblems by setting (n + 3.5) equal to 6.873863542 and -6.873863542.Subproblem 1
n + 3.5 = 6.873863542 Simplifying n + 3.5 = 6.873863542 Reorder the terms: 3.5 + n = 6.873863542 Solving 3.5 + n = 6.873863542 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 = 6.873863542 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 6.873863542 + -3.5 n = 6.873863542 + -3.5 Combine like terms: 6.873863542 + -3.5 = 3.373863542 n = 3.373863542 Simplifying n = 3.373863542Subproblem 2
n + 3.5 = -6.873863542 Simplifying n + 3.5 = -6.873863542 Reorder the terms: 3.5 + n = -6.873863542 Solving 3.5 + n = -6.873863542 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 = -6.873863542 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -6.873863542 + -3.5 n = -6.873863542 + -3.5 Combine like terms: -6.873863542 + -3.5 = -10.373863542 n = -10.373863542 Simplifying n = -10.373863542Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.373863542, -10.373863542}Solution
n = {3.373863542, -10.373863542}
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