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Simplifying n(n + -1) = 840 Reorder the terms: n(-1 + n) = 840 (-1 * n + n * n) = 840 (-1n + n2) = 840 Solving -1n + n2 = 840 Solving for variable 'n'. Reorder the terms: -840 + -1n + n2 = 840 + -840 Combine like terms: 840 + -840 = 0 -840 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '840' to each side of the equation. -840 + -1n + 840 + n2 = 0 + 840 Reorder the terms: -840 + 840 + -1n + n2 = 0 + 840 Combine like terms: -840 + 840 = 0 0 + -1n + n2 = 0 + 840 -1n + n2 = 0 + 840 Combine like terms: 0 + 840 = 840 -1n + n2 = 840 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. -1n + 0.25 + n2 = 840 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 840 + 0.25 Combine like terms: 840 + 0.25 = 840.25 0.25 + -1n + n2 = 840.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 840.25 Calculate the square root of the right side: 28.987066081 Break this problem into two subproblems by setting (n + -0.5) equal to 28.987066081 and -28.987066081.Subproblem 1
n + -0.5 = 28.987066081 Simplifying n + -0.5 = 28.987066081 Reorder the terms: -0.5 + n = 28.987066081 Solving -0.5 + n = 28.987066081 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 = 28.987066081 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 28.987066081 + 0.5 n = 28.987066081 + 0.5 Combine like terms: 28.987066081 + 0.5 = 29.487066081 n = 29.487066081 Simplifying n = 29.487066081Subproblem 2
n + -0.5 = -28.987066081 Simplifying n + -0.5 = -28.987066081 Reorder the terms: -0.5 + n = -28.987066081 Solving -0.5 + n = -28.987066081 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 = -28.987066081 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -28.987066081 + 0.5 n = -28.987066081 + 0.5 Combine like terms: -28.987066081 + 0.5 = -28.487066081 n = -28.487066081 Simplifying n = -28.487066081Solution
The solution to the problem is based on the solutions from the subproblems. n = {29.487066081, -28.487066081}
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