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Simplifying 10n2 + 35 = 65n Reorder the terms: 35 + 10n2 = 65n Solving 35 + 10n2 = 65n Solving for variable 'n'. Reorder the terms: 35 + -65n + 10n2 = 65n + -65n Combine like terms: 65n + -65n = 0 35 + -65n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(7 + -13n + 2n2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(7 + -13n + 2n2)' equal to zero and attempt to solve: Simplifying 7 + -13n + 2n2 = 0 Solving 7 + -13n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + -6.5n + n2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + -6.5n + -3.5 + n2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + -6.5n + n2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + -6.5n + n2 = 0 + -3.5 -6.5n + n2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 -6.5n + n2 = -3.5 The n term is -6.5n. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5n + 10.5625 + n2 = -3.5 + 10.5625 Reorder the terms: 10.5625 + -6.5n + n2 = -3.5 + 10.5625 Combine like terms: -3.5 + 10.5625 = 7.0625 10.5625 + -6.5n + n2 = 7.0625 Factor a perfect square on the left side: (n + -3.25)(n + -3.25) = 7.0625 Calculate the square root of the right side: 2.657536453 Break this problem into two subproblems by setting (n + -3.25) equal to 2.657536453 and -2.657536453.Subproblem 1
n + -3.25 = 2.657536453 Simplifying n + -3.25 = 2.657536453 Reorder the terms: -3.25 + n = 2.657536453 Solving -3.25 + n = 2.657536453 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = 2.657536453 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = 2.657536453 + 3.25 n = 2.657536453 + 3.25 Combine like terms: 2.657536453 + 3.25 = 5.907536453 n = 5.907536453 Simplifying n = 5.907536453Subproblem 2
n + -3.25 = -2.657536453 Simplifying n + -3.25 = -2.657536453 Reorder the terms: -3.25 + n = -2.657536453 Solving -3.25 + n = -2.657536453 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + n = -2.657536453 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + n = -2.657536453 + 3.25 n = -2.657536453 + 3.25 Combine like terms: -2.657536453 + 3.25 = 0.592463547 n = 0.592463547 Simplifying n = 0.592463547Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.907536453, 0.592463547}Solution
n = {5.907536453, 0.592463547}
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