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Simplifying 10n2 + -35n = 65 Reorder the terms: -35n + 10n2 = 65 Solving -35n + 10n2 = 65 Solving for variable 'n'. Reorder the terms: -65 + -35n + 10n2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -35n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-13 + -7n + 2n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-13 + -7n + 2n2)' equal to zero and attempt to solve: Simplifying -13 + -7n + 2n2 = 0 Solving -13 + -7n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + -3.5n + n2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + -3.5n + 6.5 + n2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + -3.5n + n2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + -3.5n + n2 = 0 + 6.5 -3.5n + n2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 -3.5n + n2 = 6.5 The n term is -3.5n. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5n + 3.0625 + n2 = 6.5 + 3.0625 Reorder the terms: 3.0625 + -3.5n + n2 = 6.5 + 3.0625 Combine like terms: 6.5 + 3.0625 = 9.5625 3.0625 + -3.5n + n2 = 9.5625 Factor a perfect square on the left side: (n + -1.75)(n + -1.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (n + -1.75) equal to 3.092329219 and -3.092329219.Subproblem 1
n + -1.75 = 3.092329219 Simplifying n + -1.75 = 3.092329219 Reorder the terms: -1.75 + n = 3.092329219 Solving -1.75 + n = 3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = 3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = 3.092329219 + 1.75 n = 3.092329219 + 1.75 Combine like terms: 3.092329219 + 1.75 = 4.842329219 n = 4.842329219 Simplifying n = 4.842329219Subproblem 2
n + -1.75 = -3.092329219 Simplifying n + -1.75 = -3.092329219 Reorder the terms: -1.75 + n = -3.092329219 Solving -1.75 + n = -3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + n = -3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + n = -3.092329219 + 1.75 n = -3.092329219 + 1.75 Combine like terms: -3.092329219 + 1.75 = -1.342329219 n = -1.342329219 Simplifying n = -1.342329219Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.842329219, -1.342329219}Solution
n = {4.842329219, -1.342329219}
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