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Simplifying 4n(n + 10) = 34 Reorder the terms: 4n(10 + n) = 34 (10 * 4n + n * 4n) = 34 (40n + 4n2) = 34 Solving 40n + 4n2 = 34 Solving for variable 'n'. Reorder the terms: -34 + 40n + 4n2 = 34 + -34 Combine like terms: 34 + -34 = 0 -34 + 40n + 4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-17 + 20n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-17 + 20n + 2n2)' equal to zero and attempt to solve: Simplifying -17 + 20n + 2n2 = 0 Solving -17 + 20n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8.5 + 10n + n2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + 10n + 8.5 + n2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + 10n + n2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + 10n + n2 = 0 + 8.5 10n + n2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 10n + n2 = 8.5 The n term is 10n. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10n + 25 + n2 = 8.5 + 25 Reorder the terms: 25 + 10n + n2 = 8.5 + 25 Combine like terms: 8.5 + 25 = 33.5 25 + 10n + n2 = 33.5 Factor a perfect square on the left side: (n + 5)(n + 5) = 33.5 Calculate the square root of the right side: 5.787918451 Break this problem into two subproblems by setting (n + 5) equal to 5.787918451 and -5.787918451.Subproblem 1
n + 5 = 5.787918451 Simplifying n + 5 = 5.787918451 Reorder the terms: 5 + n = 5.787918451 Solving 5 + n = 5.787918451 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 5.787918451 + -5 Combine like terms: 5 + -5 = 0 0 + n = 5.787918451 + -5 n = 5.787918451 + -5 Combine like terms: 5.787918451 + -5 = 0.787918451 n = 0.787918451 Simplifying n = 0.787918451Subproblem 2
n + 5 = -5.787918451 Simplifying n + 5 = -5.787918451 Reorder the terms: 5 + n = -5.787918451 Solving 5 + n = -5.787918451 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -5.787918451 + -5 Combine like terms: 5 + -5 = 0 0 + n = -5.787918451 + -5 n = -5.787918451 + -5 Combine like terms: -5.787918451 + -5 = -10.787918451 n = -10.787918451 Simplifying n = -10.787918451Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.787918451, -10.787918451}Solution
n = {0.787918451, -10.787918451}
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