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Simplifying n(n + 2) = 9.8596 Reorder the terms: n(2 + n) = 9.8596 (2 * n + n * n) = 9.8596 (2n + n2) = 9.8596 Solving 2n + n2 = 9.8596 Solving for variable 'n'. Reorder the terms: -9.8596 + 2n + n2 = 9.8596 + -9.8596 Combine like terms: 9.8596 + -9.8596 = 0.0000 -9.8596 + 2n + n2 = 0.0000 Begin completing the square. Move the constant term to the right: Add '9.8596' to each side of the equation. -9.8596 + 2n + 9.8596 + n2 = 0.0000 + 9.8596 Reorder the terms: -9.8596 + 9.8596 + 2n + n2 = 0.0000 + 9.8596 Combine like terms: -9.8596 + 9.8596 = 0.0000 0.0000 + 2n + n2 = 0.0000 + 9.8596 2n + n2 = 0.0000 + 9.8596 Combine like terms: 0.0000 + 9.8596 = 9.8596 2n + n2 = 9.8596 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 9.8596 + 1 Reorder the terms: 1 + 2n + n2 = 9.8596 + 1 Combine like terms: 9.8596 + 1 = 10.8596 1 + 2n + n2 = 10.8596 Factor a perfect square on the left side: (n + 1)(n + 1) = 10.8596 Calculate the square root of the right side: 3.29539072 Break this problem into two subproblems by setting (n + 1) equal to 3.29539072 and -3.29539072.Subproblem 1
n + 1 = 3.29539072 Simplifying n + 1 = 3.29539072 Reorder the terms: 1 + n = 3.29539072 Solving 1 + n = 3.29539072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 3.29539072 + -1 Combine like terms: 1 + -1 = 0 0 + n = 3.29539072 + -1 n = 3.29539072 + -1 Combine like terms: 3.29539072 + -1 = 2.29539072 n = 2.29539072 Simplifying n = 2.29539072Subproblem 2
n + 1 = -3.29539072 Simplifying n + 1 = -3.29539072 Reorder the terms: 1 + n = -3.29539072 Solving 1 + n = -3.29539072 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -3.29539072 + -1 Combine like terms: 1 + -1 = 0 0 + n = -3.29539072 + -1 n = -3.29539072 + -1 Combine like terms: -3.29539072 + -1 = -4.29539072 n = -4.29539072 Simplifying n = -4.29539072Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.29539072, -4.29539072}
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