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Simplifying n(n + 1) = 22 * 22 * 2 Reorder the terms: n(1 + n) = 22 * 22 * 2 (1 * n + n * n) = 22 * 22 * 2 (1n + n2) = 22 * 22 * 2 Multiply 22 * 22 1n + n2 = 484 * 2 Multiply 484 * 2 1n + n2 = 968 Solving 1n + n2 = 968 Solving for variable 'n'. Reorder the terms: -968 + 1n + n2 = 968 + -968 Combine like terms: 968 + -968 = 0 -968 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '968' to each side of the equation. -968 + 1n + 968 + n2 = 0 + 968 Reorder the terms: -968 + 968 + 1n + n2 = 0 + 968 Combine like terms: -968 + 968 = 0 0 + 1n + n2 = 0 + 968 1n + n2 = 0 + 968 Combine like terms: 0 + 968 = 968 1n + n2 = 968 The n term is 1n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 968 + 0.25 Reorder the terms: 0.25 + n + n2 = 968 + 0.25 Combine like terms: 968 + 0.25 = 968.25 0.25 + n + n2 = 968.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 968.25 Calculate the square root of the right side: 31.116715765 Break this problem into two subproblems by setting (n + 0.5) equal to 31.116715765 and -31.116715765.Subproblem 1
n + 0.5 = 31.116715765 Simplifying n + 0.5 = 31.116715765 Reorder the terms: 0.5 + n = 31.116715765 Solving 0.5 + n = 31.116715765 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 31.116715765 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 31.116715765 + -0.5 n = 31.116715765 + -0.5 Combine like terms: 31.116715765 + -0.5 = 30.616715765 n = 30.616715765 Simplifying n = 30.616715765Subproblem 2
n + 0.5 = -31.116715765 Simplifying n + 0.5 = -31.116715765 Reorder the terms: 0.5 + n = -31.116715765 Solving 0.5 + n = -31.116715765 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -31.116715765 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -31.116715765 + -0.5 n = -31.116715765 + -0.5 Combine like terms: -31.116715765 + -0.5 = -31.616715765 n = -31.616715765 Simplifying n = -31.616715765Solution
The solution to the problem is based on the solutions from the subproblems. n = {30.616715765, -31.616715765}
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