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Simplifying 5n2 + 10n + -85 = 0 Reorder the terms: -85 + 10n + 5n2 = 0 Solving -85 + 10n + 5n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '5'. 5(-17 + 2n + n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-17 + 2n + n2)' equal to zero and attempt to solve: Simplifying -17 + 2n + n2 = 0 Solving -17 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '17' to each side of the equation. -17 + 2n + 17 + n2 = 0 + 17 Reorder the terms: -17 + 17 + 2n + n2 = 0 + 17 Combine like terms: -17 + 17 = 0 0 + 2n + n2 = 0 + 17 2n + n2 = 0 + 17 Combine like terms: 0 + 17 = 17 2n + n2 = 17 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 17 + 1 Reorder the terms: 1 + 2n + n2 = 17 + 1 Combine like terms: 17 + 1 = 18 1 + 2n + n2 = 18 Factor a perfect square on the left side: (n + 1)(n + 1) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (n + 1) equal to 4.242640687 and -4.242640687.Subproblem 1
n + 1 = 4.242640687 Simplifying n + 1 = 4.242640687 Reorder the terms: 1 + n = 4.242640687 Solving 1 + n = 4.242640687 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 4.242640687 + -1 Combine like terms: 1 + -1 = 0 0 + n = 4.242640687 + -1 n = 4.242640687 + -1 Combine like terms: 4.242640687 + -1 = 3.242640687 n = 3.242640687 Simplifying n = 3.242640687Subproblem 2
n + 1 = -4.242640687 Simplifying n + 1 = -4.242640687 Reorder the terms: 1 + n = -4.242640687 Solving 1 + n = -4.242640687 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -4.242640687 + -1 Combine like terms: 1 + -1 = 0 0 + n = -4.242640687 + -1 n = -4.242640687 + -1 Combine like terms: -4.242640687 + -1 = -5.242640687 n = -5.242640687 Simplifying n = -5.242640687Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.242640687, -5.242640687}Solution
n = {3.242640687, -5.242640687}
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