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Simplifying n(n + 2) = 32 Reorder the terms: n(2 + n) = 32 (2 * n + n * n) = 32 (2n + n2) = 32 Solving 2n + n2 = 32 Solving for variable 'n'. Reorder the terms: -32 + 2n + n2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + 2n + 32 + n2 = 0 + 32 Reorder the terms: -32 + 32 + 2n + n2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 2n + n2 = 0 + 32 2n + n2 = 0 + 32 Combine like terms: 0 + 32 = 32 2n + n2 = 32 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 = 32 + 1 Reorder the terms: 1 + 2n + n2 = 32 + 1 Combine like terms: 32 + 1 = 33 1 + 2n + n2 = 33 Factor a perfect square on the left side: (n + 1)(n + 1) = 33 Calculate the square root of the right side: 5.744562647 Break this problem into two subproblems by setting (n + 1) equal to 5.744562647 and -5.744562647.Subproblem 1
n + 1 = 5.744562647 Simplifying n + 1 = 5.744562647 Reorder the terms: 1 + n = 5.744562647 Solving 1 + n = 5.744562647 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 = 5.744562647 + -1 Combine like terms: 1 + -1 = 0 0 + n = 5.744562647 + -1 n = 5.744562647 + -1 Combine like terms: 5.744562647 + -1 = 4.744562647 n = 4.744562647 Simplifying n = 4.744562647Subproblem 2
n + 1 = -5.744562647 Simplifying n + 1 = -5.744562647 Reorder the terms: 1 + n = -5.744562647 Solving 1 + n = -5.744562647 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 = -5.744562647 + -1 Combine like terms: 1 + -1 = 0 0 + n = -5.744562647 + -1 n = -5.744562647 + -1 Combine like terms: -5.744562647 + -1 = -6.744562647 n = -6.744562647 Simplifying n = -6.744562647Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.744562647, -6.744562647}
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