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Simplifying 10n2 + 130n + -345 = 0 Reorder the terms: -345 + 130n + 10n2 = 0 Solving -345 + 130n + 10n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '5'. 5(-69 + 26n + 2n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-69 + 26n + 2n2)' equal to zero and attempt to solve: Simplifying -69 + 26n + 2n2 = 0 Solving -69 + 26n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -34.5 + 13n + n2 = 0 Move the constant term to the right: Add '34.5' to each side of the equation. -34.5 + 13n + 34.5 + n2 = 0 + 34.5 Reorder the terms: -34.5 + 34.5 + 13n + n2 = 0 + 34.5 Combine like terms: -34.5 + 34.5 = 0.0 0.0 + 13n + n2 = 0 + 34.5 13n + n2 = 0 + 34.5 Combine like terms: 0 + 34.5 = 34.5 13n + n2 = 34.5 The n term is 13n. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13n + 42.25 + n2 = 34.5 + 42.25 Reorder the terms: 42.25 + 13n + n2 = 34.5 + 42.25 Combine like terms: 34.5 + 42.25 = 76.75 42.25 + 13n + n2 = 76.75 Factor a perfect square on the left side: (n + 6.5)(n + 6.5) = 76.75 Calculate the square root of the right side: 8.760707734 Break this problem into two subproblems by setting (n + 6.5) equal to 8.760707734 and -8.760707734.Subproblem 1
n + 6.5 = 8.760707734 Simplifying n + 6.5 = 8.760707734 Reorder the terms: 6.5 + n = 8.760707734 Solving 6.5 + n = 8.760707734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 8.760707734 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 8.760707734 + -6.5 n = 8.760707734 + -6.5 Combine like terms: 8.760707734 + -6.5 = 2.260707734 n = 2.260707734 Simplifying n = 2.260707734Subproblem 2
n + 6.5 = -8.760707734 Simplifying n + 6.5 = -8.760707734 Reorder the terms: 6.5 + n = -8.760707734 Solving 6.5 + n = -8.760707734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -8.760707734 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -8.760707734 + -6.5 n = -8.760707734 + -6.5 Combine like terms: -8.760707734 + -6.5 = -15.260707734 n = -15.260707734 Simplifying n = -15.260707734Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.260707734, -15.260707734}Solution
n = {2.260707734, -15.260707734}
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