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Simplifying n(3n + -39) = 1 Reorder the terms: n(-39 + 3n) = 1 (-39 * n + 3n * n) = 1 (-39n + 3n2) = 1 Solving -39n + 3n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + -39n + 3n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -39n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -13n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -13n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -13n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -13n + n2 = 0 + 0.3333333333 -13n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -13n + n2 = 0.3333333333 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 = 0.3333333333 + 42.25 Reorder the terms: 42.25 + -13n + n2 = 0.3333333333 + 42.25 Combine like terms: 0.3333333333 + 42.25 = 42.5833333333 42.25 + -13n + n2 = 42.5833333333 Factor a perfect square on the left side: (n + -6.5)(n + -6.5) = 42.5833333333 Calculate the square root of the right side: 6.52559065 Break this problem into two subproblems by setting (n + -6.5) equal to 6.52559065 and -6.52559065.Subproblem 1
n + -6.5 = 6.52559065 Simplifying n + -6.5 = 6.52559065 Reorder the terms: -6.5 + n = 6.52559065 Solving -6.5 + n = 6.52559065 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 = 6.52559065 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = 6.52559065 + 6.5 n = 6.52559065 + 6.5 Combine like terms: 6.52559065 + 6.5 = 13.02559065 n = 13.02559065 Simplifying n = 13.02559065Subproblem 2
n + -6.5 = -6.52559065 Simplifying n + -6.5 = -6.52559065 Reorder the terms: -6.5 + n = -6.52559065 Solving -6.5 + n = -6.52559065 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 = -6.52559065 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + n = -6.52559065 + 6.5 n = -6.52559065 + 6.5 Combine like terms: -6.52559065 + 6.5 = -0.02559065 n = -0.02559065 Simplifying n = -0.02559065Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.02559065, -0.02559065}
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