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Simplifying n(n + 1) = 168 Reorder the terms: n(1 + n) = 168 (1 * n + n * n) = 168 (1n + n2) = 168 Solving 1n + n2 = 168 Solving for variable 'n'. Reorder the terms: -168 + 1n + n2 = 168 + -168 Combine like terms: 168 + -168 = 0 -168 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '168' to each side of the equation. -168 + 1n + 168 + n2 = 0 + 168 Reorder the terms: -168 + 168 + 1n + n2 = 0 + 168 Combine like terms: -168 + 168 = 0 0 + 1n + n2 = 0 + 168 1n + n2 = 0 + 168 Combine like terms: 0 + 168 = 168 1n + n2 = 168 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 = 168 + 0.25 Reorder the terms: 0.25 + n + n2 = 168 + 0.25 Combine like terms: 168 + 0.25 = 168.25 0.25 + n + n2 = 168.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 168.25 Calculate the square root of the right side: 12.971121771 Break this problem into two subproblems by setting (n + 0.5) equal to 12.971121771 and -12.971121771.Subproblem 1
n + 0.5 = 12.971121771 Simplifying n + 0.5 = 12.971121771 Reorder the terms: 0.5 + n = 12.971121771 Solving 0.5 + n = 12.971121771 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 = 12.971121771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 12.971121771 + -0.5 n = 12.971121771 + -0.5 Combine like terms: 12.971121771 + -0.5 = 12.471121771 n = 12.471121771 Simplifying n = 12.471121771Subproblem 2
n + 0.5 = -12.971121771 Simplifying n + 0.5 = -12.971121771 Reorder the terms: 0.5 + n = -12.971121771 Solving 0.5 + n = -12.971121771 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 = -12.971121771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -12.971121771 + -0.5 n = -12.971121771 + -0.5 Combine like terms: -12.971121771 + -0.5 = -13.471121771 n = -13.471121771 Simplifying n = -13.471121771Solution
The solution to the problem is based on the solutions from the subproblems. n = {12.471121771, -13.471121771}
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