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Simplifying n(n + 2) = 2.2201 Reorder the terms: n(2 + n) = 2.2201 (2 * n + n * n) = 2.2201 (2n + n2) = 2.2201 Solving 2n + n2 = 2.2201 Solving for variable 'n'. Reorder the terms: -2.2201 + 2n + n2 = 2.2201 + -2.2201 Combine like terms: 2.2201 + -2.2201 = 0.0000 -2.2201 + 2n + n2 = 0.0000 Begin completing the square. Move the constant term to the right: Add '2.2201' to each side of the equation. -2.2201 + 2n + 2.2201 + n2 = 0.0000 + 2.2201 Reorder the terms: -2.2201 + 2.2201 + 2n + n2 = 0.0000 + 2.2201 Combine like terms: -2.2201 + 2.2201 = 0.0000 0.0000 + 2n + n2 = 0.0000 + 2.2201 2n + n2 = 0.0000 + 2.2201 Combine like terms: 0.0000 + 2.2201 = 2.2201 2n + n2 = 2.2201 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 = 2.2201 + 1 Reorder the terms: 1 + 2n + n2 = 2.2201 + 1 Combine like terms: 2.2201 + 1 = 3.2201 1 + 2n + n2 = 3.2201 Factor a perfect square on the left side: (n + 1)(n + 1) = 3.2201 Calculate the square root of the right side: 1.794463708 Break this problem into two subproblems by setting (n + 1) equal to 1.794463708 and -1.794463708.Subproblem 1
n + 1 = 1.794463708 Simplifying n + 1 = 1.794463708 Reorder the terms: 1 + n = 1.794463708 Solving 1 + n = 1.794463708 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 = 1.794463708 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.794463708 + -1 n = 1.794463708 + -1 Combine like terms: 1.794463708 + -1 = 0.794463708 n = 0.794463708 Simplifying n = 0.794463708Subproblem 2
n + 1 = -1.794463708 Simplifying n + 1 = -1.794463708 Reorder the terms: 1 + n = -1.794463708 Solving 1 + n = -1.794463708 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 = -1.794463708 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.794463708 + -1 n = -1.794463708 + -1 Combine like terms: -1.794463708 + -1 = -2.794463708 n = -2.794463708 Simplifying n = -2.794463708Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.794463708, -2.794463708}
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